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In mathematics, the conjugate transpose, Hermitian transpose, or adjoint matrix of an m-by-n matrix A with complex entries is the n-by-m matrix A^{*} obtained from A by taking the transpose and then taking the complex conjugate of each entry. The conjugate transpose is formally defined by
## Example

If
## Basic remarks

## Motivation

## Properties of the conjugate transpose

## Generalizations

## See also

## External links

- $(A^*)\_\{ij\}\; =\; overline\{A\_\{ji\}\}$

This definition can also be written as

- $A^*\; =\; (overline\{A\})^mathrm\{T\}\; =\; overline\{A^mathrm\{T\}\}$

Other names for the conjugate transpose of a matrix are Hermitian conjugate, or transjugate. The conjugate transpose of a matrix A can be denoted by any of these symbols:

- $A^*\; ,!$ or $A^mathrm\{H\}\; ,!$, commonly used in linear algebra
- $A^dagger\; ,!$, universally used in quantum mechanics
- $A^+\; ,!$, although this symbol is more commonly used for the Moore-Penrose pseudoinverse

Note that in some contexts $A^*\; ,!$ can be used to denote the matrix with complex conjugated entries so care must be taken not to confuse notations.

- $A\; =\; begin\{bmatrix\}\; 3\; +\; i\; \&\; 5\; 2-2i\; \&\; i\; end\{bmatrix\}$

- $A^*\; =\; begin\{bmatrix\}\; 3-i\; \&\; 2+2i\; 5\; \&\; -i\; end\{bmatrix\}.$

If the entries of A are real, then A^{*} coincides with the transpose A^{T} of A. It is often useful to think of square complex matrices as "generalized complex numbers", and of the conjugate transpose as a generalization of complex conjugation.

A square matrix A with entries $a\_\{ij\}$ is called

- Hermitian or self-adjoint if A = A
^{*}, i.e., $a\_\{ij\}=a\_\{ji\}^*$ ; - skew Hermitian or antihermitian if A = −A
^{*}, i.e., $a\_\{ij\}=-a\_\{ji\}^\{*\}$ ; - normal if A
^{*}A = AA^{*}.

Even if A is not square, the two matrices A^{*}A and AA^{*} are both Hermitian and in fact positive semi-definite matrices.

The adjoint matrix A^{*} should not be confused with the adjugate adj(A) (which is also sometimes called "adjoint").

The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2×2 skew-symmetric matrices, obeying matrix addition and multiplication:

- $a\; +\; ib\; equiv\; Big(begin\{matrix\}\; a\; \&\; -b\; b\; \&\; a\; end\{matrix\}Big).$

An m-by-n matrix of complex numbers could therefore equally well be represented by a 2m-by-2n matrix of real numbers. It therefore arises very naturally that when transposing such a matrix which is made up of complex numbers, one may in the process also have to take the complex conjugate of each entry.

- (A + B)
^{*}= A^{*}+ B^{*}for any two matrices A and B of the same dimensions. - (rA)
^{*}= r^{*}A^{*}for any complex number r and any matrix A. Here r^{*}refers to the complex conjugate of r. - (AB)
^{*}= B^{*}A^{*}for any m-by-n matrix A and any n-by-p matrix B. Note that the order of the factors is reversed. - (A
^{*})^{*}= A for any matrix A. - If A is a square matrix, then det(A
^{*}) = (det A)^{*}and tr(A^{*}) = (tr A)^{*} - A is invertible if and only if A
^{*}is invertible, and in that case we have (A^{*})^{−1}= (A^{−1})^{*}. - The eigenvalues of A
^{*}are the complex conjugates of the eigenvalues of A. - <Ax,y> = <x, A
^{*}y> for any m-by-n matrix A, any vector x in C^{n}and any vector y in C^{m}. Here <·,·> denotes the standard complex inner product on C^{m}and C^{n}.

The last property given above shows that if one views A as a linear transformation from the Euclidean Hilbert space C^{n} to C^{m}, then the matrix A^{*} corresponds to the adjoint operator of 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 A is a linear map from a complex vector space V to another W, then the complex conjugate linear map as well as the transposed linear map are defined, and we may thus take the conjugate transpose of A to be the complex conjugate of the transpose of A. It maps the conjugate dual of W to the conjugate dual of V.

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Last updated on Friday August 15, 2008 at 20:30:53 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Friday August 15, 2008 at 20:30:53 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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