Unitary group

2014-02-01T23:13:43Z

128.97.159.98:

In [[linear algebra]], a '''diagonal matrix''' is a [[matrix (mathematics)|matrix]] (usually a [[square matrix]]) in which the entries outside the [[main diagonal]] (↘) are all zero. The diagonal entries themselves may or may not be zero. Thus, the matrix D = (di,j) with ''n'' columns and ''n'' rows is diagonal if:

:<math>d_{i,j} = 0 \mbox{ if } i \ne j\ \forall i,j \in \{1, 2, \ldots, n\}</math>

For example, the following matrix is diagonal:
:<math>\begin{bmatrix}
1 & 0 & 0\\
0 & 4 & 0\\
0 & 0 & -3\end{bmatrix}</math>

The term ''diagonal matrix'' may sometimes refer to a '''rectangular diagonal matrix''', which is an ''m''-by-''n'' matrix with only the entries of the form ''di,i'' possibly non-zero. For example:
:<math>\begin{bmatrix}
1 & 0 & 0\\
0 & 4 & 0\\
0 & 0 & -3\\
0 & 0 & 0\\
\end{bmatrix}</math> or <math>\begin{bmatrix}
1 & 0 & 0 & 0 & 0\\
0 & 4 & 0& 0 & 0\\
0 & 0 & -3& 0 & 0\end{bmatrix}</math>

However, in the remainder of this article we will consider only square matrices. Any square diagonal matrix is also a [[symmetric matrix]]. Also, if the entries come from the [[field (mathematics)|field]] '''R''' or '''C''', then it is a [[normal matrix]] as well. Equivalently, we can define a diagonal matrix as a matrix that is both [[triangular matrix|upper-]] and [[triangular matrix|lower-triangular]]. The [[identity matrix]] ''I''''n'' and any square [[zero matrix]] are diagonal. A one-dimensional matrix is always diagonal.

== Scalar matrix ==
A diagonal matrix with all its main diagonal entries equal is a '''scalar matrix''', that is, a scalar multiple λ''I'' of the [[identity matrix]] ''I''. Its effect on a vector is [[scalar multiplication]] by λ. For example, a 3×3 scalar matrix has the form:
:<math>
\begin{bmatrix}
\lambda & 0 & 0 \\
0 & \lambda & 0 \\
0 & 0 & \lambda
\end{bmatrix} \equiv \lambda \boldsymbol{I_3}
</math>

The scalar matrices are the [[center of an algebra|center]] of the algebra of matrices: that is, they are precisely the matrices that commute with all other square matrices of the same size.

For an abstract vector space ''V'' (rather than the concrete vector space <math>K^n</math>), or more generally a [[module (ring theory)|module]] ''M'' over a [[ring (algebra)|ring]] ''R,'' with the [[endomorphism algebra]] End(''M'') (algebra of linear operators on ''M'') replacing the algebra of matrices, the analog of scalar matrices are '''scalar transformations'''. Formally, scalar multiplication is a linear map, inducing a map <math>R \to \operatorname{End}(M),</math> (send a scalar λ to the corresponding scalar transformation, multiplication by λ) exhibiting End(''M'') as a ''R''-[[Algebra (ring theory)|algebra]]. For vector spaces, or more generally [[free module]]s <math>M \cong R^n</math>, for which the endomorphism algebra is isomorphic to a matrix algebra, the scalar transforms are exactly the [[center of a ring|center]] of the endomorphism algebra, and similarly invertible transforms are the center of the [[general linear group]] GL(''V''), where they are denoted by Z(''V''), follow the usual notation for the center.

== Matrix operations ==
The operations of matrix addition and [[matrix multiplication]] are especially simple for diagonal matrices. Write diag(''a''1,...,''a''''n'') for a diagonal matrix whose diagonal entries starting in the upper left corner are ''a''1,...,''a''''n''. Then, for addition, we have

:diag(''a''1,...,''a''''n'') + diag(''b''1,...,''b''''n'') = diag(''a''1+''b''1,...,''a''''n''+''b''''n'')

and for [[matrix multiplication]],

:diag(''a''1,...,''a''''n'') · diag(''b''1,...,''b''''n'') = diag(''a''1''b''1,...,''a''''n''''b''''n'').

The diagonal matrix diag(''a''1,...,''a''''n'') is [[invertible matrix|invertible]] [[if and only if]] the entries ''a''1,...,''a''''n'' are all non-zero. In this case, we have

:diag(''a''1,...,''a''''n'')-1 = diag(''a''1-1,...,''a''''n''-1).

In particular, the diagonal matrices form a [[subring]] of the ring of all ''n''-by-''n'' matrices.

Multiplying an ''n''-by-''n'' matrix ''A'' from the ''left'' with diag(''a''1,...,''a''''n'') amounts to multiplying the ''i''-th ''row'' of ''A'' by ''a''''i'' for all ''i''; multiplying the matrix ''A'' from the ''right'' with diag(''a''1,...,''a''''n'') amounts to multiplying the ''i''-th ''column'' of ''A'' by ''a''''i'' for all ''i''.

== Operator matrix in eigenbasis ==

{{Main|Transformation_matrix#Finding the matrix of a transformation|Eigenvalues and eigenvectors|l1=Finding the matrix of a transformation}}

As explained in [[transformation_matrix#Finding_the_matrix_of_a_transformation|determining coefficients of operator matrix]], there is a special basis, ''e''1, ..., ''e''''n'', for which the matrix takes the diagonal form. Being diagonal means that all coefficients <math>a_{i,j} </math> but <math>a_{i,i}</math> are zeros in the defining equation <math>A \vec e_j = \sum a_{i,j} \vec e_i</math>, leaving only one term per sum. The surviving diagonal elements, <math>a_{i,i}</math>, are known as '''eigenvalues''' and designated with <math>\lambda_i</math> in the equation, which reduces to <math>A \vec e_i = \lambda_i \vec e_i</math>. The resulting equation is known as '''eigenvalue equation'''<ref>{{cite book |last=Nearing |first=James |year=2010 |title=Mathematical Tools for Physics |url=http://www.physics.miami.edu/nearing/mathmethods |chapter=Chapter 7.9: Eigenvalues and Eigenvectors |chapterurl= http://www.physics.miami.edu/~nearing/mathmethods/operators.pdf |accessdate=January 1, 2012|isbn=048648212X}}</ref> and used to derive the [[characteristic polynomial]] and, further, [[eigenvalues and eigenvectors]].

In other words, the [[eigenvalue]]s of diag(''λ''1, ..., ''λ''''n'') are ''λ''1, ..., ''λ''''n'' with associated [[eigenvectors]] of ''e''1, ..., ''e''''n''.

== Other properties ==

The [[determinant]] of diag(''a''1, ..., ''a''''n'') is the product ''a''1...''a''''n''.

The [[adjugate]] of a diagonal matrix is again diagonal.

A square matrix is diagonal if and only if it is triangular and [[Normal_matrix|normal]].

== Uses ==
Diagonal matrices occur in many areas of linear algebra. Because of the simple description of the matrix operation and eigenvalues/eigenvectors given above, it is always desirable to represent a given matrix or [[linear operator|linear map]] by a diagonal matrix.

In fact, a given ''n''-by-''n'' matrix ''A'' is [[similar matrix|similar]] to a diagonal matrix (meaning that there is a matrix ''X'' such that ''X-1AX'' is diagonal) if and only if it has ''n'' [[linearly independent]] eigenvectors. Such matrices are said to be [[diagonalizable matrix|diagonalizable]].

Over the [[field (mathematics)|field]] of [[real number|real]] or [[complex number|complex]] numbers, more is true. The [[spectral theorem]] says that every [[normal matrix]] is [[matrix similarity|unitarily similar]] to a diagonal matrix (if ''AA''* = ''A''*''A'' then there exists a [[unitary matrix]] ''U'' such that ''UAU''* is diagonal). Furthermore, the [[singular value decomposition]] implies that for any matrix ''A'', there exist unitary matrices ''U'' and ''V'' such that ''UAV''* is diagonal with positive entries.

== Operator theory ==
In [[operator theory]], particularly the study of [[PDEs]], operators are particularly easy to understand and PDEs easy to solve if the operator is diagonal with respect to the basis with which one is working; this corresponds to a [[separable partial differential equation]]. Therefore, a key technique to understanding operators is a change of coordinates – in the language of operators, an [[integral transform]] – which changes the basis to an [[eigenbasis]] of [[eigenfunction]]s: which makes the equation separable. An important example of this is the [[Fourier transform]], which diagonalizes constant coefficient differentiation operators (or more generally translation invariant operators), such as the Laplacian operator, say, in the [[heat equation]].

Especially easy are [[multiplication operator]]s, which are defined as multiplication by (the values of) a fixed function – the values of the function at each point correspond to the diagonal entries of a matrix.

== See also ==
{{colbegin}}
* [[Anti-diagonal matrix]]
* [[Banded matrix]]
* [[Bidiagonal matrix]]
* [[Diagonally dominant matrix]]
* [[Diagonalizable matrix]]
* [[Multiplication operator]]
* [[Tridiagonal matrix]]
* [[Toeplitz matrix]]
* [[Toral Lie algebra]]
* [[Circulant matrix]]
{{colend}}

== References ==

<references />
* Roger A. Horn and Charles R. Johnson, ''Matrix Analysis'', Cambridge University Press, 1985. ISBN 0-521-30586-1 (hardback), ISBN 0-521-38632-2 (paperback).

[[Category:Matrix normal forms]]
[[Category:Sparse matrices]]

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Unitary group