en>ZX95: removing unnecessary wikilink to belief/* World Religions */

2014-12-11T15:48:22Z

removing unnecessary wikilink to beliefWorld Religions

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	~~{{redirect\|Adjoint matrix\|~~the ~~transpose of cofactor\|Adjugate matrix}}~~		Eusebio is the name women and men use to call my home and I think it sounds quite good when you say it. I obtained to be unemployed although now I am a very [http://photo.net/gallery/tag-search/search?query_string=cashier cashier]. My house is this in South Carolina yet I don't plan directly on [http://Thesaurus.com/browse/changing changing] it. It's not a common thing but what I like doing is bottle counter tops collecting and now My partner have time to have a look at on new things. I'm not okay at webdesign but your preferred retail stores want to check our website: http://prometeu.net<br><br>
	In [~~[mathematics]], the '''conjugate transpose''', '''Hermitian transpose''', '''Hermitian conjugate''', '''bedaggered matrix''', or '''adjoint matrix''' of an ''m''~~-~~by-''n'' [[matrix (mathematics)\|matrix]~~] '~~'{{math\|A}}'' with~~ [~~[complex number\|complex]] entries is the ''n''-by-''m'' matrix ''{{math\|A}}''<sup>*<~~/~~sup> obtained from ''{{math\|A}}'' by taking the [[transpose~~]~~] and then taking the [[complex conjugate]] of each entry (i~~.~~e., negating their imaginary parts~~ but ~~not their real parts). The conjugate transpose~~ is ~~formally defined by~~
	~~:<math>(\mathbf{A}^*)_{ij} = \overline{\mathbf{A}_{ji}}</math>~~
	~~where the subscripts denote the ''i'',''j''-th entry, for 1 ≤ ''i'' ≤ ''n''~~ and ~~1 ≤ ''j'' ≤ '~~'m~~'', and the overbar denotes a scalar [[complex conjugate]]~~. ~~(The complex conjugate of~~ <~~math~~>~~a + bi~~<~~/math~~>~~, where ''a'' and ''b'' are reals, is <math>a - bi</math>.)~~

	~~This definition can also be written as~~		my website: hack clash of clans, [http://prometeu.net look at these guys],
	:~~<math>\mathbf{A}^* = (\overline{\mathbf{A}})^\mathrm{T} = \overline{\mathbf{A}^\mathrm{T}}</math>~~
	~~where <math>\mathbf{A}^\mathrm{T} \,\!</math> denotes the transpose and <math>\overline{\mathbf{A}} \,\!</math> denotes the matrix with complex conjugated entries.~~

	~~Other names for the conjugate transpose~~ of ~~a matrix are ''Hermitian conjugate''~~, ~~or '''transjugate'''. The conjugate transpose of a matrix ''{{math\|A}}'' can be denoted by any of these symbols:~~
	* <math>\mathbf{A}^* \,\!</math> or <math>\mathbf{A}^\mathrm{H} \,\!</math>, commonly used in [[~~linear algebra]]~~
	* <math>\mathbf{A}^\dagger \,\!</math> (sometimes pronounced as "''{{math\|A}}'' [[dagger (typography)\|dagger]]"), universally used in [[quantum mechanics]]
	* <math>\mathbf{A}^+ \,\!</math>, although this symbol is more commonly used for the [[Moore–Penrose pseudoinverse]]

	In some contexts, <math>\mathbf{A}^* \,\!</math> denotes the matrix with complex conjugated entries, and the conjugate transpose is then denoted by <math>\mathbf{A}^{\mathrm{T}} \,\!</math> or <math>\mathbf{A}^{\mathrm{T}} \,\!</math>.

	~~==Example==~~
	If
	:~~<math>\mathbf{A} = \begin{bmatrix} 3 + i & 5 & -2i \\ 2-2i & i & -7-13i \end{bmatrix}<~~/~~math>~~
	~~then~~
	~~:<math>\mathbf{A}^* = \begin{bmatrix} 3-i & 2+2i \\ 5 & -i \\ 2i & -7+13i\end{bmatrix}<~~/~~math>~~

	~~==Basic remarks==~~

	~~A square matrix ''{{math\|A}}'' with entries <math>a_{ij}</math> is called~~
	* [[hermitian matrix\|Hermitian]] or self-adjoint if ''{{math\|A}}'' = ''{{math\|A}}''<sup>*</sup>, i.~~e., <math>a_{ij}=\overline{a_{ji}}</math> .~~
	* [[skew-Hermitian matrix\|skew Hermitian]] or antihermitian if ''{{math\|A}}'' = −''{{math\|A}}''<sup>*</sup>, i.e., <math>a_{ij}=-\overline{a_{ji}}</math> .
	* [[normal matrix\|normal]] if ''{{math\|A}}''<sup></sup>''{{math\|A}}'' = ''{{math\|AA}}''<sup></sup>.
	* [[Unitary_matrix\|unitary]] if ''{{math\|A}}''<sup>*</sup> = ''{{math\|A}}''<sup>-1</sup>.

	Even if ''{{math\|A}}'' is not square, the two matrices ''{{math\|A}}''<sup></sup>''{{math\|A}}'' and ''{{math\|AA}}''<sup></sup> are both Hermitian and in fact [[Positive-definite matrix\|positive semi-definite matrices]].

	~~The conjugate transpose "adjoint" matrix ''{{math\|A}}''<sup>*</sup> should not be confused with the [[adjugate]] adj(''{{math\|A}}''), which is also sometimes called "adjoint".~~

	Finding the conjugate transpose of a matrix ''{{math\|A}}'' with [[Real number\|real]] entries reduces to finding the [[transpose]] of ''{{math\|A}}'', as the conjugate of a real number is the number itself.

	~~== Motivation ==~~
	~~The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2×2 real matrices, obeying matrix addition and multiplication:~~

	~~:<math>a + ib \equiv \left(\begin{matrix} a & -b \\ b & a \end{matrix}\right). </math>~~

	~~That is, denoting each ''complex'' number ''z'' by the ''real'' 2×2 matrix of the linear transformation on the [[Argand diagram~~]~~] (viewed as the ''real'' vector space <math>\mathbb{R}^2</math>) affected by complex ''z''-multiplication on <math>\mathbb{C}</math>.~~

	An ''m''-by-''n'' matrix of complex numbers could therefore equally well be represented by a ''2m''-by-''2n'' matrix of real numbers. The conjugate transpose therefore arises very naturally as the result of simply transposing such a matrix, ~~when viewed back again as ''n''-by-''m'' matrix made up of complex numbers.~~

	~~==Properties of the conjugate transpose==~~
	* (''{{math\|A}}'' + ''{{math\|B}}'')<sup></sup> = ''{{math\|A}}''<sup></sup> + ''{{math\|B}}''<sup>*</sup> for any two matrices ''{{math\|A}}'' and ''{{math\|B}}'' of the same dimensions.
	* (''r'' ''{{math\|A}}'')<sup></sup> = ''r''<sup></sup>''{{math\|A}}''<sup></sup> for any complex number ''r'' and any matrix ''{{math\|A}}''. Here ''r''<sup></sup> refers to the complex conjugate of ''r''.
	* (''{{math\|AB}}'')<sup></sup> = ''{{math\|B}}''<sup></sup>''{{math\|A}}''<sup>*</sup> for any ''m''-by-''n'' matrix ''{{math\|A}}'' and any ''n''-by-''p'' matrix ''{{math\|B}}''. Note that the order of the factors is reversed.
	* (''{{math\|A}}''<sup></sup>)<sup></sup> = ''{{math\|A}}'' for any matrix ''{{math\|A}}''.
	* If ''{{math\|A}}'' is a square matrix, then [[determinant\|det]](''{{math\|A}}''<sup></sup>) = (det ''{{math\|A}}'')<sup></sup> and [[trace (matrix)\|tr]](''{{math\|A}}''<sup></sup>) = (tr ''{{math\|A}}'')<sup></sup>
	* ''{{math\|A}}'' is [[invertible matrix\|invertible]] [[if and only if]] ''{{math\|A}}''<sup></sup> is invertible, and in that case (''{{math\|A}}''<sup></sup>)<sup>−1</sup> = (''{{math\|A}}''<sup>−1</sup>)<sup>*</sup>.
	* The [[eigenvalue]]s of ''{{math\|A}}''<sup>*</sup> are the complex conjugates of the [[eigenvalue]]s of ''{{math\|A}}''.
	* <math>\langle \mathbf{Ax}, \mathbf{y}\rangle = \langle \mathbf{x},\mathbf{A}^* \mathbf{y} \rangle</math> for any ''m''-by-''n'' matrix ''{{math\|A}}'', any vector ''x'' in <math> \mathbb{C}^n </math> and any vector ''y'' in <math> \mathbb{C}^m </math>. Here, <math>\langle\cdot,\cdot\rangle</math> denotes the standard complex [[inner product]] on <math> \mathbb{C}^m </math> and <math> \mathbb{C}^n </math>.

	~~==Generalizations==~~
	The last property given above shows that if one views ''{{math\|A}}'' as a [[linear transformation]] from Euclidean [[Hilbert space]] <math> \mathbb{C}^n </math> to <math> \mathbb{C}^m </math>, then the matrix ''{{math\|A}}''<sup>*</sup> corresponds to the [[Hermitian adjoint\|adjoint operator]] of ''{{math\|A}}''. The concept of adjoint operators between Hilbert spaces can thus be seen as a generalization of the conjugate transpose of matrices.

	Another generalization is available: suppose ''{{math\|A}}'' is a linear map from a complex [[vector space]] ''{{math\|V}}'' to another, ''{{math\|W}}'', then the [[complex conjugate linear map]] as well as the [[transpose of a linear map\|transposed linear map]] are defined, and we may thus take the conjugate transpose of ''{{math\|A}}'' to be the complex conjugate of the transpose of ''{{math\|A}}''. It maps the conjugate [[dual space\|dual]] of ''{{math\|W}}'' to the conjugate dual of ''{{math\|V}}''.

	~~==See also==~~
	*[[Hermitian adjoint\|Hermitian conjugate]]
	*[[Adjugate matrix]]

	~~==External links==~~
	* {{springer\|title=Adjoint matrix\|id=p/a010850}}
	* {{MathWorld \| urlname=ConjugateTranspose \| title=Conjugate Transpose}}
	* {{planetmath reference\|id=4382\|title=Conjugate transpose}}

	~~[[Category:Linear algebra]]~~
	~~[[Category:Matrices]]~~

	~~[[ja:随伴行列]]~~
	~~[[pl:Macierz sprzężona hermitowsko]]~~
	~~[[uk:Спряжена матриця]]~~

en>Delusion23: fix typos, dates, formatting, references, brackets and links, replaced: Metric Time → Metric time using AWB

2014-01-05T23:39:39Z

fix typos, dates, formatting, references, brackets and links, replaced: Metric Time → Metric time using AWB

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en>Melab-1 at 01:47, 28 August 2012

2012-08-28T01:47:45Z

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en>ZX95: removing unnecessary wikilink to belief/* World Religions */

en>Delusion23: fix typos, dates, formatting, references, brackets and links, replaced: Metric Time → Metric time using AWB

en>Melab-1 at 01:47, 28 August 2012