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In [[mathematics]], the '''matrix exponential''' is a [[matrix function]] on [[square matrix|square matrices]] analogous to the ordinary [[exponential function]]. Abstractly, the matrix exponential gives the connection between a matrix [[Lie algebra]] and the corresponding [[Lie group]].
In [[computational geometry]], '''polygon triangulation''' is the decomposition of a [[polygonal area]] ([[simple polygon]]) '''P''' into a set of [[triangles]],<ref name= bkos>{{Citation|author = [[Mark de Berg]], [[Marc van Kreveld]], [[Mark Overmars]], and [[Otfried Schwarzkopf]] | year = 2000 | title = Computational Geometry | publisher = [[Springer-Verlag]] | edition = 2nd revised | isbn = 3-540-65620-0}} Chapter 3: Polygon Triangulation: pp.45–61.</ref> i.e., finding the set of triangles with pairwise non-intersecting interiors whose union is '''P'''.


Let {{mvar|X}}  be an {{math|''n''×''n''}} [[real number|real]] or [[complex number|complex]] [[matrix (mathematics)|matrix]]. The exponential of {{mvar|X}}, denoted by {{math|''e''<sup>''X''</sup>}} or {{math|exp(''X'')}}, is the {{math|''n''×''n''}} matrix given by the [[power series]]
Triangulations may be viewed as special cases of [[planar straight-line graph]]s. When there are no holes or added points, triangulations form [[outerplanar graph|maximal outerplanar graphs]].


:<math>e^X = \sum_{k=0}^\infty{1 \over k!}X^k.</math>
== Polygon triangulation without extra vertices ==


The above series always converges, so the exponential of {{mvar|X}} is well-defined. Note that if {{mvar|X}} is a 1×1 matrix the matrix exponential of {{mvar|X}} is a 1×1 matrix whose single element is the ordinary [[Exponential function|exponential]] of the single element of {{mvar|X}}.
Over time a number of algorithms have been proposed to triangulate a polygon.


==Properties==
=== Special cases ===
Let {{math|''X''}} and {{math|''Y''}} be {{math|''n''×''n''}} complex matrices and let {{math|''a''}} and {{math|''b''}} be arbitrary complex numbers. We denote the {{math|''n''×''n''}} [[identity matrix]] by {{math|''I''}} and the [[zero matrix]] by 0. The matrix exponential satisfies the following properties:
A [[convex polygon]] is trivial to triangulate in [[linear time]], by adding diagonals from one vertex to all other vertices. The total number of ways to triangulate a convex ''n''-gon by non-intersecting diagonals is the (''n''&nbsp;&minus;&nbsp;2)-th [[Catalan number]], which equals <math>\tfrac{n \cdot (n+1) \cdots (2n-4)}{(n-2)!}</math>, a solution found by [[Leonhard Euler]].<ref>[[Clifford Pickover|Pickover, Clifford A.]], ''The Math Book'', Sterling, 2009: p. 184.</ref>


* {{math|''e''<sup>0</sup> {{=}} ''I''}}
A [[monotone polygon]] can be triangulated in linear time with either the algorithm of  [[Alain Fournier|A. Fournier]] and D.Y. Montuno,<ref>{{Citation |last1=Fournier |first1=A. |author1-link=Alain Fournier |last2=Montuno |first2=D. Y. |author2-link= |title=Triangulating simple polygons and equivalent problems |journal=[[ACM Transactions on Graphics]] |volume=3 |issue=2 | year=1984 <!--|month=April--> |pages=153–174 |issn=0730-0301 |doi=10.1145/357337.357341}}</ref> or the algorithm of [[Godfried Toussaint]].<ref>Toussaint, Godfried T. (1984), "A new linear algorithm for triangulating monotone polygons," ''Pattern Recognition Letters'', '''2''' (March):155–158.</ref>
* {{math|''e<sup>''aX''</sup>''e''<sup>''bX''</sup> {{=}} ''e''<sup>(''a'' + ''b'')''X''</sup>}}
* {{math|''e''<sup>''X''</sup>''e''<sup>&minus;''X''</sup> {{=}} ''I''}}
* If {{math|''XY'' {{=}} ''YX''}} then {{math|''e''<sup>''X''</sup>''e''<sup>''Y''</sup> {{=}} ''e''<sup>''Y''</sup>''e''<sup>''X''</sup> {{=}} ''e''<sup>(''X''&nbsp;+&nbsp;''Y'')</sup>.}}
* If {{math|''Y''}} is [[invertible matrix|invertible]] then {{math|''e''<sup>''YXY''<sup>&minus;1</sup></sup> {{=}}''Ye''<sup>''X''</sup>''Y''<sup>&minus;1</sup>.}}
* {{math|exp(''X''<sup>T</sup>) {{=}} (exp ''X'')<sup>T</sup>}}, where {{math|''X''<sup>T</sup>}} denotes the [[transpose]] of {{math|''X''}}. It follows that if {{math|''X''}} is [[symmetric matrix|symmetric]] then {{math|''e''<sup>''X''</sup>}} is also symmetric, and that if {{math|''X''}} is [[skew-symmetric matrix|skew-symmetric]] then {{math|''e''<sup>''X''</sup>}} is [[orthogonal matrix|orthogonal]].
* {{math|exp(''X''<sup>*</sup>) {{=}} (exp ''X'')<sup>*</sup>}}, where {{math|''X''<sup>*</sup>}} denotes the [[conjugate transpose]] of {{math|''X''}}. It follows that if {{math|''X''}} is [[Hermitian matrix|Hermitian]] then {{math|''e''<sup>''X''</sup>}} is also Hermitian, and that if {{math|''X''}} is [[skew-Hermitian matrix|skew-Hermitian]] then {{math|''e''<sup>''X''</sup>}} is [[unitary matrix|unitary]].


===Linear differential equation systems===
===Ear clipping method===
[[Image:Polygon-ear.png|thumb|A polygon ear]]
One way to triangulate a simple polygon is based on the fact that any simple polygon with at least 4 vertices without holes has at least two '[[Ear (mathematics)|ear]]s', which are triangles with two sides being the edges of the polygon and the third one completely inside it (and with an extra property unimportant for triangulation).<ref>Meisters, G. H., "Polygons have ears." American Mathematical Monthly 82 (1975). 648–651</ref> The algorithm then consists of finding such an ear, removing it from the polygon (which results in a new polygon that still meets the conditions) and repeating until there is only one triangle left.


{{main|matrix differential equation}}
This algorithm is easy to implement, but slower than some other algorithms, and it only works on polygons without holes. An implementation that keeps separate lists of convex and concave vertices will run in ''O''(''n''<sup>2</sup>) time. This method is known as ''ear clipping'' and sometimes ''ear trimming''. An efficient algorithm for cutting off ears was discovered by Hossam ElGindy, Hazel Everett, and [[Godfried Toussaint]].<ref>ElGindy, H., Everett, H., and Toussaint, G. T., (1993) "Slicing an ear using prune-and-search," ''Pattern Recognition Letters'', '''14''', (9):719–722.</ref>


One of the reasons for the importance of the matrix exponential is that it can be used to solve systems of linear [[ordinary differential equations]]. The solution of
===Using monotone polygons===
: <math> \frac{d}{dt} y(t) = Ay(t), \quad y(0) = y_0, </math>
[[Image:Polygon-to-monotone.png|thumb|Breaking a polygon into monotone polygons]]
where {{mvar|A}} is a constant matrix, is given by
A simple polygon may be decomposed into [[monotone polygon]]s as follows.<ref name= bkos/>
: <math> y(t) = e^{At} y_0. \, </math>
The matrix exponential can also be used to solve the inhomogeneous equation
: <math> \frac{d}{dt} y(t) = Ay(t) + z(t), \quad y(0) = y_0. </math>
See the [[#Applications|section on applications below]] for examples.


There is no closed-form solution for differential equations of the form
For each point, check if the neighboring points are both on the same side of the '[[sweep line]]', a horizontal or vertical line on which the point being iterated lies. If they are, check the next sweep line on the other side.  Break the polygon on the line between the original point and one of the points on this one.
: <math> \frac{d}{dt} y(t) = A(t) \, y(t), \quad y(0) = y_0, </math>
where {{mvar|A}}  is not constant, but the [[Magnus series]] gives the solution as an infinite sum.


===The exponential of sums===
Note that if you are moving downwards, the points where both of the vertices are below the sweep line are 'split points'. They mark a split in the polygon. From there you have to consider both sides separately.


We know that the exponential function satisfies {{math|''e''<sup>''x''+''y''</sup> {{=}} ''e''<sup>''x''</sup> ''e''<sup>''y''</sup>}} for any real numbers (scalars) {{mvar|x}} and {{mvar|y}}. The same goes for commuting matrices: If the matrices {{mvar|X}} and {{mvar|Y}} commute (meaning that {{math|''XY'' {{=}} ''YX''}}), then
Using this algorithm to triangulate a simple polygon takes ''O''(''n''&nbsp;log&nbsp;''n'') time.


:<math>e^{X+Y} = e^Xe^Y ~.</math>
=== Dual graph of a triangulation ===


However, if they do not commute, then the above equality does not necessarily hold, in which case the [[Baker–Campbell–Hausdorff formula]] furnishes {{math|''e''<sup>''X''+''Y''</sup>}}.
A useful graph that is often associated with a triangulation of a polygon {{math|<var>P</var>}} is the [[dual graph]]. Given a triangulation {{math|<var>T<sub>P</sub></var>}} of {{math|<var>P</var>}}, one defines the graph {{math|<var>G</var>(<var>T<sub>P</sub></var>)}} as the graph whose vertex set are the triangles of {{math|<var>T<sub>P</sub></var>}}, two vertices (triangles) being adjacent if and only if they share a diagonal. It is easy to observe that {{math|<var>G</var>(<var>T<sub>P</sub></var>)}} is a [[Tree (graph theory)|tree]] with maximum degree 3.


The converse is false: the equation  {{math|''e''<sup>''X''+''Y''</sup> {{=}} ''e''<sup>''X''</sup> ''e''<sup>''Y''</sup>}} does not necessarily imply that {{mvar|X}} and {{mvar|Y}} commute.  
===Computational complexity===
For a long time, there was an open problem in computational geometry whether a [[simple polygon]] can be triangulated faster than {{math|O(<var>n</var> log <var>n</var>)}} time.<ref name= bkos/> Then, {{harvtxt|Tarjan|Van Wyk|1988}} discovered an {{math|O(<var>n</var> log log <var>n</var>)}}-time algorithm for triangulation,<ref>{{citation
| last1 = Tarjan | first1 = Robert E. | author1-link = Robert Tarjan
| last2 = Van Wyk | first2 = Christopher J.
| doi = 10.1137/0217010
| issue = 1
| journal = [[SIAM Journal on Computing]]
| mr = 925194
| pages = 143–178
| title = An O(''n'' log log ''n'')-time algorithm for triangulating a simple polygon
| volume = 17
| year = 1988}}.</ref> later simplified by {{harvtxt|Kirkpatrick|Klawe|Tarjan|1992}}.<ref>{{citation
| last1 = Kirkpatrick | first1 = David G. | author1-link = David G. Kirkpatrick
| last2 = Klawe | first2 = Maria M. | author2-link = Maria Klawe
| last3 = Tarjan | first3 = Robert E. | author3-link = Robert Tarjan
| doi = 10.1007/BF02187846
| issue = 4
| journal = [[Discrete and Computational Geometry]]
| mr = 1148949
| pages = 329–346
| title = Polygon triangulation in O(''n'' log log ''n'') time with simple data structures
| volume = 7
| year = 1992}}.</ref> Several improved methods with complexity [[Big O notation#Orders of common functions|{{math|O(<var>n</var> log<sup>*</sup> <var>n</var>)}}]] (in practice, indistinguishable from [[linear time]]) followed.<ref>{{citation
| last1 = Clarkson | first1 = Kenneth L. | author1-link = Kenneth L. Clarkson
| last2 = Tarjan | first2 = Robert | author2-link = Robert Tarjan
| last3 = van Wyk | first3 = Christopher J.
| doi = 10.1007/BF02187741
| journal = [[Discrete and Computational Geometry]]
| pages = 423–432
| title = A fast Las Vegas algorithm for triangulating a simple polygon
| volume = 4
| year = 1989}}.</ref><ref>{{Citation |last=Seidel|first=Raimund |author-link= Raimund Seidel| title=A Simple and Fast Incremental Randomized Algorithm for Computing Trapezoidal Decompositions and for Triangulating Polygons |journal=Computational Geometry: Theory and Applications |volume=1 |year=1991 |pages=51–64}}</ref><ref>{{citation
| last1 = Clarkson | first1 = Kenneth L. | author1-link = Kenneth L. Clarkson
| last2 = Cole | first2 = Richard
| last3 = Tarjan | first3 = Robert E. | author3-link = Robert Tarjan
| doi = 10.1142/S0218195992000081
| issue = 2
| journal = International Journal of Computational Geometry & Applications
| mr = 1168952
| pages = 117–133
| title = Randomized parallel algorithms for trapezoidal diagrams
| volume = 2
| year = 1992}}.</ref>


For [[Hermitian matrix| Hermitian matrices]] there are two notable theorems related to the [[Matrix trace|trace]] of matrix exponentials.
[[Bernard Chazelle]] showed in 1991 that any simple polygon can be triangulated in linear time, though the proposed algorithm is very complex.<ref>{{Citation |last=Chazelle |first=Bernard |author-link=Bernard Chazelle | title=Triangulating a Simple Polygon in Linear Time |journal=Discrete &amp; Computational Geometry |volume=6 |year=1991|pages=485–524 |issn=0179-5376 |doi=10.1007/BF02574703}}</ref> A simpler randomized algorithm with linear expected time is also known.<ref>{{Citation |last1=Amato |first1=Nancy M. |last2=Goodrich |first2=Michael T. |author2-link=Michael T. Goodrich|last3=Ramos |first3=Edgar A. |title=A Randomized Algorithm for Triangulating a Simple Polygon in Linear Time |journal=Discrete &amp; Computational Geometry |volume=26 |year=2001 <!--|month=May--> |pages=245–265 |issn=0179-5376 |doi=10.1007/s00454-001-0027-x |url=http://parasol.tamu.edu/publications/abstract.php?pub_id=185 |issue=2}}</ref>
====Golden–Thompson inequality====
{{main|Golden–Thompson inequality}}


If {{mvar|A}}  and {{mvar|H}} are Hermitian matrices, then
Seidel's decomposition algorithm and Chazelle's triangulation method are discussed in detail in {{harvtxt|Li|Klette|2011}}.
:<math>\operatorname{tr}\exp(A+H) \leq \operatorname{tr}(\exp(A)\exp(H)). </math> <ref>{{cite book | author=Bhatia, R. | title=Matrix Analysis |series=Graduate Texts in Mathematics|isbn=978-0-387-94846-1 | year = 1997 | publisher=Springer | volume=169}}</ref>
<ref>{{citation
Note that there is no requirement of commutativity.  There are counterexamples to show that the Golden–Thompson inequality cannot be extended to three matrices−−and, in any event, {{math|tr(exp(''A'')exp(''B'')exp(''C''))}} is not guaranteed to be real for Hermitian {{math|''A'' , ''B'', ''C''}}.  However, the next theorem accomplishes this in a way.
| last1 = Li | first1 = Fajie
| last2 = Klette | first2 = Reinhard
| title = Euclidean Shortest Paths
| publisher = [[Springer (publisher)|Springer]]
| doi = 10.1007/978-1-4471-2256-2
| ISBN = 978-1-4471-2255-5
| year = 2011}}.</ref>


====Lieb's theorem====
The [[time complexity]] of triangulation of an {{math|<var>n</var>}}-vertex polygon ''with'' holes has an {{math|Ω(<var>n</var> log <var>n</var>)}} [[lower bound]].<ref name= bkos/>
The '''Lieb's theorem''', named after [[Elliott H. Lieb]], states that, for a fixed [[Hermitian matrix]] {{mvar|H}},  the function
:<math> f(A) = \operatorname{tr} \,\exp \left (H + \log A \right) </math>
is [[Concave function | concave]] on the [[Convex cone | cone]] of [[positive-definite matrix | positive-definite matrices]]. <ref>{{cite journal|doi=10.1016/0001-8708(73)90011-X | author = E. H. Lieb | title=Convex trace functions and the Wigner–Yanase–Dyson conjecture | journal=Adv. Math. | volume=11 | page=267–288 | year=1973|issue=3|ref=harv}}
{{cite journal|doi=10.1007/BF01646492 | author=H. Epstein | title=Remarks on two theorems of E. Lieb | journal=Commun Math. Phys. |volume=31|page=317–325 | year=1973|issue=4|ref=harv}}</ref>


===The exponential map===
== See also ==
* [[Nonzero-rule]]
* [[Tessellation]]
* [[Catalan number]]
* [[Point set triangulation]]
* [[Delaunay triangulation]]
* [[Tiling by regular polygons]]
* [[Minimum-weight triangulation]], for a point set and for a simple polygon
* [[Planar graph]]
* [[Polygon covering#Covering a polygon with triangles]]


Note that the exponential of a matrix is always an [[invertible matrix]]. The inverse matrix of {{math|''e''<sup>''X''</sup>}} is given by {{math|''e''<sup>&minus;''X''</sup>}}. This is analogous to the fact that the exponential of a complex number is always nonzero. The matrix exponential then gives us a map
== References ==
:<math>\exp \colon M_n(\mathbb C) \to \mathrm{GL}(n,\mathbb C)</math>
{{reflist}}
from the space of all ''n''×''n'' matrices to the [[general linear group]] of degree {{mvar|n}}, i.e. the [[group (mathematics)|group]] of all ''n''×''n'' invertible matrices. In fact, this map is [[surjective]] which means that every invertible matrix can be written as the exponential of some other matrix (for this, it is essential to consider the field '''C''' of complex numbers and not '''R''').
 
For any two matrices {{mvar|X}} and {{mvar|Y}},
:<math> \| e^{X+Y} - e^X \| \le \|Y\| e^{\|X\|} e^{\|Y\|}, </math>
 
where ||&nbsp;·&nbsp;|| denotes an arbitrary [[matrix norm]]. It follows that the exponential map is [[continuity (mathematics)|continuous]] and [[Lipschitz continuous]] on [[compact set|compact]] subsets of {{math|''M''<sub>''n''</sub>('''C''')}}.
 
The map
:<math>t \mapsto e^{tX}, \qquad t \in \mathbb R</math>
defines a [[Smooth function#Smoothness|smooth]] curve in the general linear group which passes through the identity element at ''t'' = 0.
 
In fact, this gives a [[one-parameter subgroup]] of the general linear group since
:<math>e^{tX}e^{sX} = e^{(t+s)X}.\,</math>
 
The derivative of this curve (or [[tangent vector]]) at a point ''t'' is given by
:<math>\frac{d}{dt}e^{tX} = Xe^{tX} = e^{tX}X. \qquad (1)</math>
The derivative at ''t'' = 0 is just the matrix ''X'', which is to say that ''X'' generates this one-parameter subgroup.
 
More generally,<ref>{{cite journal|doi=10.1063/1.1705306 | author = R. M. Wilcox | title=Exponential Operators and Parameter Differentiation in Quantum Physics | journal=Journal of Mathematical Physics | volume=8 | page=962–982 | year=1967|ref=harv|issue=4}}</ref> for a generic {{mvar|t}}-dependent exponent, {{math|''X(t)''}},
{{Equation box 1
|indent =:
|equation = <math>\frac{d}{dt}e^{X(t)} = \int_0^1 e^{\alpha X(t)} \frac{dX(t)}{dt} e^{(1-\alpha) X(t)}\,d\alpha ~.  </math> 
|cellpadding= 
|border
|border colour = #0073CF
|bgcolor=#F9FFF7}}
 
Taking the above expression {{math|''e''<sup>''X''(''t'')</sup>}} outside the integral sign and expanding the integrand with the help of  the [[Baker–Campbell–Hausdorff formula|Hadamard lemma]] one can obtain the following useful expression for the derivative of the matrix exponent,
:<math>\left(\frac{d}{dt}e^{X(t)}\right)e^{-X(t)} = \frac{d}{dt}X(t) + \frac{1}{2!}[X(t),\frac{d}{dt}X(t)] + \frac{1}{3!}[X(t),[X(t),\frac{d}{dt}X(t)]]+\cdots </math>
 
===The determinant of the matrix exponential===
 
By [[Jacobi's formula]], for any complex square matrix the following identity holds:
{{Equation box 1
|indent =:
|equation = <math> \det (e^A)= e^{\operatorname{tr}(A)}~.</math>
|cellpadding= 6
|border
|border colour = #0073CF
|bgcolor=#F9FFF7}}
 
In addition to providing a computational tool, this formula demonstrates that a matrix exponential is always an [[invertible matrix]]. This follows from the fact
the right hand side of the above equation is always non-zero, and so {{math|det(''e<sup>A</sup>'')≠ 0}},  which means that {{math|''e<sup>A</sup>''}} must be invertible.
 
In the real-valued case, the formula also exhibits the map
:<math>\exp \colon M_n(\mathbb R) \to \mathrm{GL}(n,\mathbb R)</math>
to not be [[surjective function|surjective]],  in contrast to  the complex case mentioned earlier. This follows from the fact that, for real-valued matrices, the right-hand side of the formula  is always positive, while there exist invertible matrices with a negative determinant.
 
==Computing the matrix exponential==
 
Finding reliable and accurate methods to compute the matrix exponential is difficult,  and this is still a topic of considerable current research in mathematics and numerical analysis. Both [[Matlab]] and [[GNU Octave]] use [[Padé approximant]].<ref>{{cite web|url=http://www.mathworks.de/help/techdoc/ref/expm.html |title=Matrix exponential - MATLAB expm - MathWorks Deutschland |publisher=Mathworks.de |date=2011-04-30 |accessdate=2013-06-05}}</ref><ref>{{cite web|url=http://www.network-theory.co.uk/docs/octave3/octave_200.html |title=GNU Octave - Functions of a Matrix |publisher=Network-theory.co.uk |date=2007-01-11 |accessdate=2013-06-05}}</ref> Several methods are listed below.
 
===Diagonalizable case===
 
If a matrix is [[diagonal matrix|diagonal]]:
 
:<math>A=\begin{bmatrix} a_1 & 0 & \ldots & 0 \\
0 & a_2 & \ldots & 0  \\ \vdots & \vdots & \ddots & \vdots \\
0 & 0 & \ldots & a_n \end{bmatrix} </math>,
 
then its exponential can be obtained by just exponentiating every entry on the main diagonal:
 
:<math>e^A=\begin{bmatrix} e^{a_1} & 0 & \ldots & 0 \\
0 & e^{a_2} & \ldots & 0  \\ \vdots & \vdots & \ddots & \vdots \\
0 & 0 & \ldots & e^{a_n} \end{bmatrix} </math>.
 
This also allows one to exponentiate [[diagonalizable matrix|diagonalizable matrices]]. If {{math|''A'' {{=}} ''UDU''<sup>&minus;1</sup>}} and {{mvar|D}} is diagonal, then {{math|''e''<sup>''A''</sup> {{=}} ''Ue''<sup>''D''</sup>''U''<sup>&minus;1</sup>}}. Application of [[Sylvester's formula]] yields the same result.  The proof behind this is that multiplication between diagonal matrices is equivalent to element wise multiplication; in particular, the "one-dimensional" exponentiation is felt element wise for the diagonal case.
===Projection case===
If the matrix under question is a [[projection matrix]] (idempotent), then the matrix exponential of it is {{math|''e''<sup>''P''</sup> {{=}} I + (''e'' &minus; 1)''P''}},  which is easy to show upon expansion of the definition of the exponential,
:<math>e^P = I + \sum_{k=1}^{\infty} \frac{P^k}{k!}=I+\left(\sum_{k=1}^{\infty} \frac{1}{k!}\right)P=I+(e-1)P      ~.</math>
 
===Nilpotent case===
 
A matrix ''N'' is [[nilpotent matrix|nilpotent]] if ''N''<sup>''q''</sup> = 0 for some integer ''q''. In this case, the matrix exponential ''e''<sup>''N''</sup> can be computed directly from the series expansion, as the series terminates after a finite number of terms:
 
:<math>e^N = I + N + \frac{1}{2}N^2 + \frac{1}{6}N^3 + \cdots + \frac{1}{(q-1)!}N^{q-1} ~.</math>
 
===Generalization===
When the [[Minimal polynomial (linear algebra)|minimal polynomial]] of a matrix ''X'' can be factored into a product of first degree polynomials, it can be expressed as a sum
:<math>X = A + N \,</math>
where
*''A'' is diagonalizable
*''N'' is nilpotent
*''A'' commutes with ''N'' (i.e. ''AN'' = ''NA'')
This is the [[Jordan–Chevalley decomposition]].
 
This means that we can compute the exponential of ''X'' by reducing to the previous two cases:
:<math>e^X = e^{A+N} = e^A e^N. \,</math>
Note that we need the commutativity of ''A'' and ''N'' for the last step to work.
 
Another (closely related) method if the field is [[algebraically closed]] is to work with the [[Jordan form]] of ''X''. Suppose that ''X'' = ''PJP''<sup>&nbsp;&minus;1</sup> where ''J'' is the Jordan form of ''X''. Then
 
:<math>e^{X}=Pe^{J}P^{-1}.\,</math>
 
Also, since
: <math>J=J_{a_1}(\lambda_1)\oplus J_{a_2}(\lambda_2)\oplus\cdots\oplus J_{a_n}(\lambda_n),</math>
 
: <math>
\begin{align}
e^{J} & {} = \exp \big( J_{a_1}(\lambda_1)\oplus J_{a_2}(\lambda_2)\oplus\cdots\oplus J_{a_n}(\lambda_n) \big) \\
& {} = \exp \big( J_{a_1}(\lambda_1) \big) \oplus \exp \big( J_{a_2}(\lambda_2) \big) \oplus\cdots\oplus \exp \big( J_{a_k}(\lambda_k) \big).
\end{align}
</math>
 
Therefore, we need only know how to compute the matrix exponential of a Jordan block. But each Jordan block is of the form
:<math>J_{a}(\lambda) = \lambda I + N \,</math>
where ''N'' is a special nilpotent matrix. The matrix exponential of this block is given by
:<math>e^{\lambda I + N} = e^{\lambda}e^N. \,</math>
 
===Evaluation by Laurent series===
By virtue of the [[Cayley–Hamilton theorem]] the matrix exponential is expressible as a polynomial of order {{mvar|n}}−1.
If {{mvar|P}} and {{math|''Q<sub>t</sub>''}} are nonzero polynomials in one variable, such that {{math|''P''(''A'') {{=}} 0}}, and if the [[meromorphic function]]
:<math>f(z)=\frac{e^{t z}-Q_t(z)}{P(z)}</math>
is [[entire function|entire]], then
:<math>e^{t A} = Q_t(A)</math>.
To prove this, multiply the first of the two above equalities by {{math|''P''(''z'')}} and replace {{mvar|z}} by {{mvar|A}}.
 
Such a polynomial {{math|''Q<sub>t</sub>(z)''}} can be found as follows−−see [[Sylvester's formula]]. Letting {{mvar|a}} be a root of {{mvar|P}},  {{math|''Q<sub>a,t</sub>(z)''}} is solved from the product of {{mvar|P}} by the [[Laurent series#Principal part|principal part]] of the [[Laurent series]] of {{mvar|f}} at {{mvar|a}}: It is proportional to the relevant [[Frobenius covariant]]. Then the sum ''S<sub>t</sub>'' of the ''Q<sub>a,t</sub>'', where {{mvar|a}} runs over all the roots of {{mvar|P}}, can be taken as a particular {{math|''Q<sub>t</sub>''}}. All the other ''Q<sub>t</sub>'' will be obtained by adding a multiple of {{mvar|P}} to {{math|''S<sub>t</sub>(z)''}}. In particular, {{math|''S<sub>t</sub>(z)''}}, the [[Sylvester's formula|Lagrange-Sylvester polynomial]], is the only {{math|''Q<sub>t</sub>''}} whose degree is less than that of {{mvar|P}}.
 
'''Example''': Consider the case of an arbitrary  2-by-2 matrix,
:<math>A:=\begin{bmatrix}
a & b \\
c & d \end{bmatrix}.</math>
 
The exponential matrix {{math|e<sup>''tA''</sup>}}, by virtue of the [[Cayley–Hamilton theorem]], must be of  the form
::<math>e^{tA}=s_0(t)\,I+s_1(t)\,A</math>.
(For any complex number {{mvar|z}} and any '''''C'''''-algebra {{mvar|B}}, we denote again by {{mvar|z}}  the product of {{mvar|z}} by the unit of {{mvar|B}}.) Let {{mvar|α}} and {{mvar|β}}  be the roots of the [[characteristic polynomial]] of {{mvar|A}},
:<math>P(z)=z^2-(a+d)\ z+ ad-bc= (z-\alpha)(z-\beta) ~ .</math>
 
Then we have
:<math>S_t(z)= e^{\alpha t} \frac{z-\beta}{\alpha-\beta}  + e^{\beta t} \frac{z-\alpha}{\beta-\alpha}  ~, </math>
and hence
:<math>s_0(t)=\frac{\alpha\,e^{\beta t}
-\beta\,e^{\alpha t}}{\alpha-\beta},\quad
s_1(t)=\frac{e^{\alpha t}-e^{\beta t}}{\alpha-\beta}\quad</math>
if  {{math|''α'' ≠ ''β''}}; while, if  {{math|''α'' {{=}} ''β''}},
:<math>S_t(z)= e^{\alpha t} ( 1+ t (z-\alpha  ))  ~, </math>
so that
:<math>s_0(t)=(1-\alpha\,t)\,e^{\alpha t},\quad
s_1(t)=t\,e^{\alpha t}~.</math>
 
Defining
:<math>s \equiv \frac{\alpha + \beta}{2}=\frac{\operatorname{tr} A}{2}~, \qquad \qquad  q\equiv \frac{\alpha-\beta}{2}=\pm\sqrt{-\det\left(A-s I\right)},</math>
we have
:<math>s_0(t) = e^{s t}\left(\cosh (q t) - s \frac{\sinh (q t)}{q}\right),\qquad s_1(t) =e^{s t}\frac{\sinh(q t)}{q},</math>
where {{math| sin(''qt'')/''q''}}  is 0 if {{mvar|t}} = 0, and {{mvar|t}} if {{mvar|q}} = 0.
Thus,
{{Equation box 1
|indent =:
|equation = <math>e^{tA}=e^{s t}\left( (\cosh (q t) - s \frac{\sinh (q t)}{q})~I~+\frac{\sinh(q t)}{q} A\right) ~.</math>
|cellpadding= 6
|border
|border colour = #0073CF
|bgcolor=#F9FFF7}}
Thus, as indicated above, the matrix {{mvar|A}} having decomposed into the sum of two mutually commuting pieces, the traceful piece and the traceless piece,
:<math> A= sI + (A-sI)~,</math>
the matrix exponential reduces to a plain product of the exponentials of the two respective pieces. This is a formula often used in physics, as it amounts to the analog of  [[Euler's formula]] for  [[Pauli_spin_matrices#Exponential_of_a_Pauli_vector|Pauli spin matrices]], that is rotations of the doublet representation of the group [[SU(2)]].
 
 
The polynomial {{math|''S<sub>t</sub>''}} can also be given the following "[[interpolation]]" characterization. Define  {{math|''e<sub>t</sub>(z) ≡ e<sup>tz</sup>''}}, and {{mvar|n}} ≡ deg{{mvar|P}}. Then  {{math|''S<sub>t</sub>(z)''}} is the unique degree {{math|< ''n''}}  polynomial which satisfies {{math|''S<sub>t</sub><sup>(k)</sup>(a)'' {{=}} ''e<sub>t</sub><sup>(k)</sup>(a)''}} whenever {{mvar|k}} is less than the multiplicity of {{mvar|a}} as a root of {{mvar|P}}.  We assume, as we obviously can, that {{mvar|P}} is the [[Minimal polynomial (linear algebra)|minimal polynomial]] of {{mvar|A}}. We further assume that {{mvar|A}} is a [[diagonalizable matrix]]. In particular, the roots of {{mvar|P}} are simple, and the "[[interpolation]]" characterization indicates that {{math|''S<sub>t</sub>''}} is given by the [[Lagrange interpolation]] formula, so it is the  [[Sylvester's formula|Lagrange−Sylvester polynomial]] .
 
At the other extreme, if {{math| ''P'' {{=}}  ''(z−a)<sup>n</sup>''}}, then
:<math>S_t=e^{at}\ \sum_{k=0}^{n-1}\ \frac{t^k}{k!}\ (z-a)^k ~.</math>
The simplest case not covered by the above observations is when <math>P=(z-a)^2\,(z-b)</math> with  {{math|''a'' ≠ ''b''}}, which yields
:<math>S_t=e^{at}\ \frac{z-b}{a-b}\ \Bigg(1+\left(t+\frac{1}{b-a}\right)(z-a)\Bigg)+e^{bt}\ \frac{(z-a)^2}{(b-a)^2}\quad.</math>
 
=== Evaluation by implementation of [[Sylvester's formula]]===
A practical, expedited computation of the above reduces to the following rapid steps.
Recall from above that an ''n''-by-''n'' matrix  {{math|exp(''tA'')}} amounts to a linear combination of the first {{mvar|n}}−1 powers of {{mvar|A}}  by the [[Cayley-Hamilton theorem]].  For [[diagonalizable matrix|diagonalizable]] matrices, as illustrated above, e.g. in the 2 by 2 case,  [[Sylvester's formula]] yields  {{math|exp(''tA'') {{=}} ''B<sub>α</sub>'' exp(''tα'')+''B<sub>β</sub>'' exp(''tβ'')}}, where the {{mvar|B}}s are  the [[Frobenius covariant]]s of {{mvar|A}}.  It is easiest, however, to simply solve for these {{mvar|B}}s directly, by evaluating this expression and its first derivative at {{mvar|t}}=0, in terms of {{mvar|A}} and {{mvar|I}},  to find  the same answer as above.
 
But this simple procedure also works for [[defective matrix|defective]] matrices, in a generalization due to Buchheim.<ref>Rinehart, R. F. (1955). "The equivalence of definitions of a matric function". ''The American Mathematical Monthly'', '''62''' (6), 395-414.</ref> This is illustrated here for a 4-by-4 example of a matrix which is not diagonalizable, and the {{mvar|B}}s are not projection matrices.
 
Consider
:<math>A =
\begin{pmatrix}
1 & 1 & 0 & 0 \\
0 & 1 & 1 & 0 \\
0 & 0 & 1 & -1/8 \\
0 & 0 & 1/2 & 1/2
\end{pmatrix}  ~,
</math>
with eigenvalues {{math| ''λ''<sub>1</sub>{{=}}3/4}}  and  {{math| ''λ''<sub>2</sub>{{=}}1}}, each with a
multiplicity of two.
 
Consider the exponential of each eigenvalue multiplied by {{mvar|t}},  {{math|exp(''λ<sub>i</sub>t'')}}. Multiply each such by the corresponding undetermined coefficient matrix {{math|''B''<sub>''i''</sub>}}.  If the eigenvalues have an algebraic multiplicity greater than 1, then repeat the process,  but now multiplying by an extra factor of {{mvar|t}}  for each repetition, to ensure linear independence. (If one eigenvalue had a multiplicity of three, then there would be the three terms: <math>B_{i_1} e^{\lambda_i t}, ~ B_{i_2} t e^{\lambda_i t}, ~ B_{i_3} t^2 e^{\lambda_i t} </math>. By contrast, when all eigenvalues are distinct, the {{mvar|B}}s are just the [[Frobenius covariant]]s, and solving for them as below just amounts to the inversion of the [[Vandermonde matrix]] of these 4 eigenvalues.) 
 
Sum all such terms,  here four such:
:<math>
e^{A t} = B_{1_1} e^{\lambda_1 t} + B_{1_2} t e^{\lambda_1 t} + B_{2_1} e^{\lambda_2 t} + B_{2_2} t e^{\lambda_2 t} ,
</math>
:<math>
e^{A t} = B_{1_1} e^{3/4 t} + B_{1_2} t e^{3/4 t} + B_{2_1} e^{1 t} + B_{2_2} t e^{1 t}
</math>.
 
To solve for all of the unknown matrices  {{mvar|B}}  in terms of the first three powers of  {{mvar|A}} and the identity, we need four equations, the above one providing one such at {{mvar|t}} =0. Further, differentiate it with respect to {{mvar|t}},
:<math>
A e^{A t} = 3/4 B_{1_1} e^{3/4 t} + \left( 3/4 t + 1 \right) B_{1_2} e^{3/4 t} + 1 B_{2_1} e^{1 t} + \left(1 t + 1 \right) B_{2_2} e^{1 t}  ~,
</math>
and again,
:<math>
\begin{align}
A^2 e^{A t} =& (3/4)^2 B_{1_1} e^{3/4 t} + \left( (3/4)^2 t + ( 3/4 + 1 \cdot 3/4) \right) B_{1_2} e^{3/4 t} + B_{2_1} e^{1 t}\\ +& \left(1^2 t + (1 + 1 \cdot 1 )\right) B_{2_2} e^{1 t} \\  =& (3/4)^2 B_{1_1} e^{3/4 t} + \left( (3/4)^2 t + 3/2 \right) B_{1_2} e^{3/4 t} + B_{2_1} e^{t} + \left(t + 2\right) B_{2_2} e^{t} ~,
\end{align}
</math>
and once more,
:<math>
\begin{align}
A^3 e^{A t} =& (3/4)^3 B_{1_1} e^{3/4 t} + \left( (3/4)^3 t + ( (3/4)^2 + (3/2) \cdot 3/4) ) \right) B_{1_2} e^{3/4 t}\\ +& B_{2_1} e^{1 t} + \left(1^3 t + (1 + 2) \cdot 1 \right) B_{2_2} e^{1 t} \\ =&  (3/4)^3 B_{1_1} e^{3/4 t}\! + \left( (3/4)^3 t\! + 27/16 ) \right) B_{1_2} e^{3/4 t}\! + B_{2_1} e^{t}\! + \left(t + 3\cdot 1\right) B_{2_2} e^{t}
\end{align} 
</math>.
(In the general case, {{mvar|n}}−1 derivatives need be taken.)
 
Setting {{mvar|t}}=0 in these four equations, the  four coefficient matrices  {{mvar|B}}s may be solved for,
:<math>
\begin{align}
I =& B_{1_1} + B_{2_1} \\
A =& 3/4 B_{1_1} + B_{1_2} + B_{2_1} + B_{2_2} \\
A^2 =& (3/4)^2 B_{1_1} + (3/2) B_{1_2} + B_{2_1} + 2 B_{2_2} \\
A^3 =& (3/4)^3 B_{1_1} + (27/16) B_{1_2} + B_{2_1} + 3 B_{2_2}
\end{align} </math> ,
to yield 
:<math>
\begin{align}
B_{1_1} =& 128 A^3 - 366 A^2 + 288 A - 80 I \\
B_{1_2} =& 16 A^3 - 44 A^2 + 40 A - 12 I \\
B_{2_1} =&-128 A^3 + 366 A^2 - 288 A + 80 I\\
B_{2_2} =& 16 A^3 - 40 A^2 + 33 A - 9 I
\end{align}
</math> .
 
Substituting with  the value for {{mvar|A}} yields the coefficient matrices
:<math>
\begin{align}
B_{1_1} =& \begin{pmatrix}0 & 0 & 48 & -16\\ 0 & 0 & -8 & 2\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{pmatrix}\\
B_{1_2} =& \begin{pmatrix}0 & 0 & 4 & -2\\ 0 & 0 & -1 & 1/2\\ 0 & 0 & 1/4 & -1/8\\ 0 & 0 & 1/2 & -1/4 \end{pmatrix}\\
B_{2_1} =& \begin{pmatrix}1 & 0 & -48 & 16\\ 0 & 1 & 8 & -2\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\end{pmatrix}\\
B_{2_2} =& \begin{pmatrix}0 & 1 & 8 & -2\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\end{pmatrix} 
\end{align}
</math>
so the final answer is
:<math>
{e}^{tA}\!=\!\begin{pmatrix}{e}^{t} & t{e}^{t} & \left( 8t-48\right) {e}^{t}\!+\left( 4t+48\right){e}^{3t/4} & \left( 16-2\,t\right){e}^{t}\!+\left( -2t-16\right){e}^{3t/4}\\ 0 & {e}^{t} & 8{e}^{t}\!+\left( -t-8\right) {e}^{3t/4} & -\frac{4{e}^{t}+\left(-t-4\right){e}^{3t/4}}{2}\\ 0 & 0 & \frac{\left( t+4\right) {e}^{3t/4}}{4} & -\frac{t {e}^{3t/4}}{8}\\ 0 & 0 & \frac{t{e}^{3t/4}}{2} & -\frac{\left( t-4\right) {e}^{3t/4}}{4}\end{pmatrix}
</math>.
 
The procedure is quite shorter than [[Matrix_differential_equation#Putzer Algorithm for computing eAt|Putzer's algorithm]] sometimes utilized in such cases.
 
==Illustrations==
 
Suppose that we want to compute the exponential of
 
: <math>B=\begin{bmatrix}
21 & 17 & 6 \\
-5 & -1 & -6 \\
4 & 4 & 16 \end{bmatrix}.</math>
 
Its Jordan form is
 
: <math>J = P^{-1}BP = \begin{bmatrix}
4 & 0 & 0 \\
0 & 16 & 1 \\
0 & 0 & 16 \end{bmatrix},</math>
 
where the matrix ''P'' is given by
 
: <math>P=\begin{bmatrix}
-\frac14 & 2 & \frac54 \\
\frac14 & -2 & -\frac14 \\
0 & 4 & 0 \end{bmatrix}.</math>
 
Let us first calculate exp(''J''). We have
 
: <math>J=J_1(4)\oplus J_2(16) \, </math>
 
The exponential of a 1×1 matrix is just the exponential of the one entry of the matrix, so exp(''J''<sub>1</sub>(4)) =&nbsp;[''e''<sup>4</sup>]. The exponential of ''J''<sub>2</sub>(16) can be calculated by the formula ''e''<sup>(λ''I''&nbsp;+&nbsp;''N'')</sup> =&nbsp;''e''<sup>λ</sup> ''e''<sup>N</sup> mentioned above; this yields<ref>This can be generalized; in general, the exponential of ''J''<sub>''n''</sub>(''a'') is an upper triangular matrix with ''e''<sup>''a''</sup>/0! on the main diagonal, ''e''<sup>''a''</sup>/1! on the one above, ''e''<sup>''a''</sup>/2! on the next one, and so on.</ref>
 
: <math>
\begin{align}
\exp \left( \begin{bmatrix} 16 & 1 \\ 0 & 16 \end{bmatrix} \right)
& = e^{16} \exp \left( \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \right) \\[6pt]
& = e^{16} \left(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} + {1 \over 2!}\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} + \cdots \right)
= \begin{bmatrix} e^{16} & e^{16} \\ 0 & e^{16} \end{bmatrix}.
\end{align}
</math>
 
Therefore, the exponential of the original matrix ''B'' is
 
: <math>
\begin{align}
\exp(B)
& = P \exp(J) P^{-1}
= P \begin{bmatrix} e^4 & 0 & 0 \\ 0 & e^{16} & e^{16} \\ 0 & 0 & e^{16}  \end{bmatrix} P^{-1} \\[6pt]
& = {1\over 4} \begin{bmatrix}
  13e^{16} - e^4 & 13e^{16} - 5e^4 & 2e^{16} - 2e^4 \\
  -9e^{16} + e^4 & -9e^{16} + 5e^4 & -2e^{16} + 2e^4 \\
  16e^{16}      & 16e^{16}        & 4e^{16}
\end{bmatrix}.
\end{align}
</math>
 
==Applications==
===Linear differential equations===
 
The matrix exponential has applications to systems of [[linear differential equation]]s.  (See also [[matrix differential equation]].)  Recall from earlier in this article that a homogeneous differential equation of the form
: <math> \mathbf{y}' = A\mathbf{y} </math>
has solution {{math|''e''<sup>''At''</sup> '''''y'''''(0)}}. If we consider the vector
: <math> \mathbf{y}(t) = \begin{pmatrix} y_1(t) \\ \vdots \\y_n(t) \end{pmatrix}  ~,</math>
we can express a system of inhomogeneous coupled linear differential equations as
: <math> \mathbf{y}'(t) = A\mathbf{y}(t)+\mathbf{b}(t).</math>
If we make an [[ansatz]] to use an integrating factor of {{math|''e''<sup>−''At''</sup>}} and multiply throughout, we obtain
: <math>e^{-At}\mathbf{y}'-e^{-At}A\mathbf{y} = e^{-At}\mathbf{b}</math>
: <math>e^{-At}\mathbf{y}'-Ae^{-At}\mathbf{y} = e^{-At}\mathbf{b}</math>
: <math> \frac{d}{dt} (e^{-At}\mathbf{y}) = e^{-At}\mathbf{b}~.</math>
 
The second step is possible due to the fact that, if {{math|''AB'' {{=}} ''BA''}}, then {{math|''e''<sup>''At''</sup>''B'' {{=}} ''Be''<sup>''At''</sup>}}. So,  calculating {{math|''e''<sup>''At''</sup>}} leads to the solution to the system, by simply integrating the third step in {{mvar|t}}s.
 
====Example (homogeneous)====
Consider  the system
:<math>\begin{matrix}
x' &=& 2x&-y&+z \\
y' &=&  &3y&-1z \\
z' &=& 2x&+y&+3z \end{matrix}</math>
 
We have the associated matrix
:<math>A=\begin{bmatrix}
2 & -1 &  1 \\
0 &  3 & -1 \\
2 &  1 &  3 \end{bmatrix}  ~.</math>
 
The matrix exponential is
:<math>e^{tA}=\begin{bmatrix}
    e^{2t}(1+e^{2t}-2t)  & -2te^{2t}    &  e^{2t}(-1+e^{2t}) \\
  -e^{2t}(-1+e^{2t}-2t) & 2(t+1)e^{2t} & -e^{2t}(-1+e^{2t}) \\
    e^{2t}(-1+e^{2t}+2t) & 2te^{2t}    &  e^{2t}(1+e^{2t})  \end{bmatrix}~,</math>
so the general solution of the homogeneous system is
: <math>\begin{bmatrix}x \\y \\ z\end{bmatrix}=
x(0)\begin{bmatrix}e^{2t}(1+e^{2t}-2t) \\-e^{2t}(-1+e^{2t}-2t)\\e^{2t}(-1+e^{2t}+2t)\end{bmatrix}
+y(0)\begin{bmatrix}-2te^{2t}\\2(t+1)e^{2t}\\2te^{2t}\end{bmatrix}
+z(0)\begin{bmatrix}e^{2t}(-1+e^{2t})\\-e^{2t}(-1+e^{2t})\\e^{2t}(1+e^{2t})\end{bmatrix} ~,</math>
amounting to
:<math>
\begin{align}
x & = x(0)(e^{2t}(1+e^{2t}-2t)) + y(0) (-2te^{2t}) + z(0)(e^{2t}(-1+e^{2t})) \\
y & = x(0)(-e^{2t}(-1+e^{2t}-2t)) + y(0)(2(t+1)e^{2t}) + z(0)(-e^{2t}(-1+e^{2t})) \\
z & = x(0)(e^{2t}(-1+e^{2t}+2t)) + y(0)(2te^{2t}) + z(0)(e^{2t}(1+e^{2t}))  ~.
\end{align}
</math>
 
====Example (inhomogeneous)====
Consider now the inhomogeneous system
:<math>\begin{matrix}
x' &=& 2x & - & y & + & z & + & e^{2t} \\
y' &=&    &  & 3y& - & z & \\
z' &=& 2x & + & y & + & 3z & + & e^{2t} \end{matrix} ~.</math>
 
We again have
:<math>A= \left[ \begin{array}{rrr}
2 & -1 &  1 \\
0 &  3 & -1 \\
2 &  1 &  3 \end{array} \right] ~,</math>
and
:<math>\mathbf{b}=e^{2t}\begin{bmatrix}1 \\0\\1\end{bmatrix}.</math>
 
From before, we already have the general solution to the homogeneous equation. Since the sum of the homogeneous and particular solutions give the general solution to the inhomogeneous problem, we now only need find the particular solution. 
 
We have, by above,
: <math>\mathbf{y}_p = e^{tA}\int_0^t e^{(-u)A}\begin{bmatrix}e^{2u} \\0\\e^{2u}\end{bmatrix}\,du+e^{tA}\mathbf{c}</math>
 
: <math>\mathbf{y}_p = e^{tA}\int_0^t
\begin{bmatrix}
    2e^u - 2ue^{2u} & -2ue^{2u}    & 0 \\  \\
-2e^u + 2(u+1)e^{2u} & 2(u+1)e^{2u} & 0 \\  \\
            2ue^{2u} & 2ue^{2u}    & 2e^u\end{bmatrix}\begin{bmatrix}e^{2u} \\0\\e^{2u}\end{bmatrix}\,du+e^{tA}\mathbf{c}</math>
 
:<math>\mathbf{y}_p = e^{tA}\int_0^t
\begin{bmatrix}
e^{2u}( 2e^u - 2ue^{2u}) \\  \\
  e^{2u}(-2e^u + 2(1 + u)e^{2u}) \\  \\
  2e^{3u} + 2ue^{4u}\end{bmatrix}+e^{tA}\mathbf{c}</math>
 
: <math>\mathbf{y}_p = e^{tA}\begin{bmatrix}
-{1 \over 24}e^{3t}(3e^t(4t-1)-16) \\  \\
{1 \over 24}e^{3t}(3e^t(4t+4)-16) \\  \\
{1 \over 24}e^{3t}(3e^t(4t-1)-16)\end{bmatrix}+
\begin{bmatrix}
    2e^t - 2te^{2t} & -2te^{2t}    & 0 \\  \\
-2e^t + 2(t+1)e^{2t} & 2(t+1)e^{2t} & 0 \\  \\
            2te^{2t} & 2te^{2t}    & 2e^t\end{bmatrix}\begin{bmatrix}c_1 \\c_2 \\c_3\end{bmatrix} ~,</math>
which could be further simplified to get the requisite particular solution determined through variation of parameters.
Note '''c''' = '''y'''<sub>''p''</sub>(0). For more rigor, see the following generalization.
 
===Inhomogeneous case generalization:  variation of parameters===
For the inhomogeneous case, we can use [[integrating factor]]s (a method akin to [[variation of parameters]]). We seek a particular solution of the form {{math|'''y'''<sub>p</sub>(''t'') {{=}} exp(''tA'')&thinsp;'''z'''&thinsp;(''t'')&thinsp;}}, 
: <math>
\begin{align}
\mathbf{y}_p'(t) & = (e^{tA})'\mathbf{z}(t)+e^{tA}\mathbf{z}'(t) \\[6pt]
& = Ae^{tA}\mathbf{z}(t)+e^{tA}\mathbf{z}'(t) \\[6pt]
& = A\mathbf{y}_p(t)+e^{tA}\mathbf{z}'(t)~.
\end{align}
</math>
 
For {{math|'''''y'''''<sub>''p''</sub>}} to be a solution,
: <math>
\begin{align}
e^{tA}\mathbf{z}'(t) & = \mathbf{b}(t) \\[6pt]
\mathbf{z}'(t) & = (e^{tA})^{-1}\mathbf{b}(t) \\[6pt]
\mathbf{z}(t) & = \int_0^t e^{-uA}\mathbf{b}(u)\,du+\mathbf{c} ~.
\end{align}
</math>
 
Thus,
: <math>
\begin{align}
\mathbf{y}_p(t) & {} = e^{tA}\int_0^t e^{-uA}\mathbf{b}(u)\,du+e^{tA}\mathbf{c} \\
& {} = \int_0^t e^{(t-u)A}\mathbf{b}(u)\,du+e^{tA}\mathbf{c}
\end{align} ~,
</math>
where {{math|'''''c'''''}} is determined by the initial conditions of the problem.
 
More precisely, consider the equation
:<math>Y'-A\ Y=F(t)</math>
with the initial condition {{math|''Y(t<sub>0</sub>)'' {{=}} ''Y<sub>0</sub>''}}, where
{{mvar|A}}  is an {{mvar|n}}  by  {{mvar|n}}  complex matrix,
 
{{mvar|F}}  is a continuous function from some open interval {{mvar|I}} to ℂ<sup>''n''</sup>,
 
<math>t_0</math> is a point of {{mvar|I}}, and
 
<math>Y_0</math> is a vector of ℂ<sup>''n''</sup>.
 
Left-multiplying the above displayed equality by {{math|''e<sup>−tA</sup>''}} yields
 
:<math>Y(t)=e^{(t-t_0)A}\ Y_0+\int_{t_0}^t e^{(t-x)A}\ F(x)\ dx  ~.</math>
 
We claim that the solution to the equation
:<math>P(d/dt)\ y = f(t)</math>
with the initial conditions <math>y^{(k)}(t_0)=y_k</math> for 0 ≤ {{math|''k < n''}} is
:<math>y(t)=\sum_{k=0}^{n-1}\ y_k\ s_k(t-t_0)+\int_{t_0}^t s_{n-1}(t-x)\ f(x)\ dx ~,</math>
where the notation is as follows:
 
<math>P\in\mathbb{C}[X]</math> is a monic polynomial of degree {{math|''n'' > 0}},
 
{{mvar|f}}  is a continuous complex valued function defined on some open interval  {{mvar|I}},
 
<math>t_0</math> is a point of  {{mvar|I}},
 
<math>y_k</math> is a complex number, and
 
{{math|''s<sub>k</sub>(t)''}}  is the coefficient of <math>X^k</math> in the polynomial denoted by <math>S_t\in\mathbb{C}[X]</math> in Subsection [[matrix exponential#Evaluation_by_Laurent_series|Evaluation by Laurent series]] above.
 
To justify this claim, we transform our order {{mvar|n}} scalar equation into an order one vector equation by the usual  [[Ordinary differential equation#Reduction to a first order system|reduction to a first order system]]. Our vector equation takes the form
 
:<math>\frac{dY}{dt}-A\ Y=F(t),\quad Y(t_0)=Y_0,</math>
 
where {{mvar|A}} is the [[transpose]] [[companion matrix]] of {{mvar|P}}. We solve this equation as explained above, computing the matrix exponentials by the observation made in Subsection [[matrix exponential#Alternative|Alternative]] above.
 
In the case {{mvar|n}} = 2  we get the following statement. The solution to
:<math>y''-(\alpha+\beta)\ y'
+\alpha\,\beta\ y=f(t),\quad
y(t_0)=y_0,\quad y'(t_0)=y_1</math>
is
:<math>y(t)=y_0\ s_0(t-t_0)+y_1\ s_1(t-t_0)
+\int_{t_0}^t s_1(t-x)\,f(x)\ dx,</math>
where the functions {{math|''s''<sub>0</sub>}}  and {{math|''s''<sub>1</sub>}} are as in  Subsection [[matrix exponential#Evaluation_by_Laurent_series|Evaluation by Laurent series]] above.
 
==See also==
{{Div col}}
*[[Matrix function]]
*[[Matrix logarithm]]
*[[Exponential function]]
*[[Exponential map]]
*[[Magnus expansion]]
*[[Vector flow]]
*[[Golden–Thompson inequality]]
*[[Phase-type distribution]]
*[[Lie product formula]]
*[[Baker–Campbell–Hausdorff formula]]
*[[Frobenius covariant]]
*[[Sylvester's formula]]
{{Div col end}}
 
==References==
{{Reflist}}
 
* {{Cite book | last1=Horn | first1=Roger A. | last2=Johnson | first2=Charles R. | title=Topics in Matrix Analysis | publisher=[[Cambridge University Press]] | isbn=978-0-521-46713-1 | year=1991 | ref=harv}}.
* {{Cite journal | last1=Moler | first1=Cleve | author1-link=Cleve Moler | last2=Van Loan | first2=Charles F. | author2-link=Charles F. Van Loan | title=Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later | year=2003 | journal=SIAM Review | issn=1095-7200 | volume=45 | issue=1 | pages=3–49 | url=http://www.cs.cornell.edu/cv/researchpdf/19ways+.pdf | doi=10.1137/S00361445024180 | ref=harv }}.


==External links==
==External links==
* {{mathworld|urlname=MatrixExponential|title=Matrix Exponential}}
* [http://computacion.cs.cinvestav.mx/~anzures/geom/triangulation.php Demo as Flash swf],  A Sweep Line algorithm.
* [http://math.fullerton.edu/mathews/n2003/MatrixExponentialMod.html Module for the Matrix Exponential]
* [http://www.songho.ca/opengl/gl_tessellation.html Song Ho's explanation of the OpenGL GLU tesselator]


{{DEFAULTSORT:Matrix Exponential}}
{{DEFAULTSORT:Polygon Triangulation}}
[[Category:Matrix theory]]
[[Category:Triangulation (geometry)]]
[[Category:Lie groups]]
[[Category:Exponentials]]

Revision as of 15:59, 12 August 2014

In computational geometry, polygon triangulation is the decomposition of a polygonal area (simple polygon) P into a set of triangles,[1] i.e., finding the set of triangles with pairwise non-intersecting interiors whose union is P.

Triangulations may be viewed as special cases of planar straight-line graphs. When there are no holes or added points, triangulations form maximal outerplanar graphs.

Polygon triangulation without extra vertices

Over time a number of algorithms have been proposed to triangulate a polygon.

Special cases

A convex polygon is trivial to triangulate in linear time, by adding diagonals from one vertex to all other vertices. The total number of ways to triangulate a convex n-gon by non-intersecting diagonals is the (n − 2)-th Catalan number, which equals , a solution found by Leonhard Euler.[2]

A monotone polygon can be triangulated in linear time with either the algorithm of A. Fournier and D.Y. Montuno,[3] or the algorithm of Godfried Toussaint.[4]

Ear clipping method

A polygon ear

One way to triangulate a simple polygon is based on the fact that any simple polygon with at least 4 vertices without holes has at least two 'ears', which are triangles with two sides being the edges of the polygon and the third one completely inside it (and with an extra property unimportant for triangulation).[5] The algorithm then consists of finding such an ear, removing it from the polygon (which results in a new polygon that still meets the conditions) and repeating until there is only one triangle left.

This algorithm is easy to implement, but slower than some other algorithms, and it only works on polygons without holes. An implementation that keeps separate lists of convex and concave vertices will run in O(n2) time. This method is known as ear clipping and sometimes ear trimming. An efficient algorithm for cutting off ears was discovered by Hossam ElGindy, Hazel Everett, and Godfried Toussaint.[6]

Using monotone polygons

Breaking a polygon into monotone polygons

A simple polygon may be decomposed into monotone polygons as follows.[1]

For each point, check if the neighboring points are both on the same side of the 'sweep line', a horizontal or vertical line on which the point being iterated lies. If they are, check the next sweep line on the other side. Break the polygon on the line between the original point and one of the points on this one.

Note that if you are moving downwards, the points where both of the vertices are below the sweep line are 'split points'. They mark a split in the polygon. From there you have to consider both sides separately.

Using this algorithm to triangulate a simple polygon takes O(n log n) time.

Dual graph of a triangulation

A useful graph that is often associated with a triangulation of a polygon Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

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A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

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This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

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Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. is a tree with maximum degree 3.

Computational complexity

For a long time, there was an open problem in computational geometry whether a simple polygon can be triangulated faster than Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. time.[1] Then, Template:Harvtxt discovered an Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.-time algorithm for triangulation,[7] later simplified by Template:Harvtxt.[8] Several improved methods with complexity Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "[http://www.blitzkappers.com/?option=com_k2&view=itemlist&task=user&id=64019 public listed property developers In singapore] agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. (in practice, indistinguishable from linear time) followed.[9][10][11]

Bernard Chazelle showed in 1991 that any simple polygon can be triangulated in linear time, though the proposed algorithm is very complex.[12] A simpler randomized algorithm with linear expected time is also known.[13]

Seidel's decomposition algorithm and Chazelle's triangulation method are discussed in detail in Template:Harvtxt. [14]

The time complexity of triangulation of an Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.-vertex polygon with holes has an Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. lower bound.[1]

See also

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

  1. ↑ 1.0 1.1 1.2 1.3 Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 Chapter 3: Polygon Triangulation: pp.45–61.
  2. ↑ Pickover, Clifford A., The Math Book, Sterling, 2009: p. 184.
  3. ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  4. ↑ Toussaint, Godfried T. (1984), "A new linear algorithm for triangulating monotone polygons," Pattern Recognition Letters, 2 (March):155–158.
  5. ↑ Meisters, G. H., "Polygons have ears." American Mathematical Monthly 82 (1975). 648–651
  6. ↑ ElGindy, H., Everett, H., and Toussaint, G. T., (1993) "Slicing an ear using prune-and-search," Pattern Recognition Letters, 14, (9):719–722.
  7. ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  8. ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  9. ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  10. ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  11. ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  12. ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  13. ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  14. ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
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