Conjugate transpose

In mathematics, the conjugate transpose, Hermitian transpose, Hermitian conjugate, bedaggered matrix, 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 (i.e., negating their imaginary parts but not their real parts). The conjugate transpose is formally defined by

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

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 $a + bi$ , where a and b are reals, is $a - bi$ .)

This definition can also be written as

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

where $\mathbf{A}^\mathrm{T} \,\!$ denotes the transpose and $\overline{\mathbf{A}} \,\!$ 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 $A$ can be denoted by any of these symbols:

$\mathbf{A}^* \,\!$ or $\mathbf{A}^\mathrm{H} \,\!$ , commonly used in linear algebra
$\mathbf{A}^\dagger \,\!$ (sometimes pronounced as " $A$ dagger"), universally used in quantum mechanics
$\mathbf{A}^+ \,\!$ , although this symbol is more commonly used for the Moore–Penrose pseudoinverse

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

1 Example
2 Basic remarks
3 Motivation
4 Properties of the conjugate transpose
5 Generalizations
6 See also
7 External links

Example []

$\mathbf{A} = \begin{bmatrix} 3 + i & 5 & -2i \\ 2-2i & i & -7-13i \end{bmatrix}$

then

$\mathbf{A}^* = \begin{bmatrix} 3-i & 2+2i \\ 5 & -i \\ 2i & -7+13i\end{bmatrix}$

Basic remarks []

A square matrix $A$ with entries $a_{ij}$ is called

Hermitian or self-adjoint if $A$ = $A$ ^*, i.e., $a_{ij}=\overline{a_{ji}}$ .
skew Hermitian or antihermitian if $A$ = − $A$ ^*, i.e., $a_{ij}=-\overline{a_{ji}}$ .
normal if $A$ ^* $A$ = $AA$ ^*.
unitary if $A$ ^* = $A$ ^-1.

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

The conjugate transpose "adjoint" matrix $A$ ^* should not be confused with the adjugate adj( $A$ ), which is also sometimes called "adjoint".

Finding the conjugate transpose of a matrix $A$ with real entries reduces to finding the transpose of $A$ , 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:

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

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 $\mathbb{R}^2$ ) affected by complex z-multiplication on $\mathbb{C}$ .

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 []

( $A$ + $B$ )^* = $A$ ^* + $B$ ^* for any two matrices $A$ and $B$ of the same dimensions.
(r $A$ )^* = 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 ( $A$ ^*)⁻¹ = ( $A$ ⁻¹)^*.
The eigenvalues of $A$ ^* are the complex conjugates of the eigenvalues of $A$ .
$\langle \mathbf{Ax}, \mathbf{y}\rangle = \langle \mathbf{x},\mathbf{A}^* \mathbf{y} \rangle$ for any m-by-n matrix $A$ , any vector x in $\mathbb{C}^n$ and any vector y in $\mathbb{C}^m$ . Here, $\langle\cdot,\cdot\rangle$ denotes the standard complex inner product on $\mathbb{C}^m$ and $\mathbb{C}^n$ .

Generalizations []

The last property given above shows that if one views $A$ as a linear transformation from Euclidean Hilbert space $\mathbb{C}^n$ to $\mathbb{C}^m$ , 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 $V$ .

External links []

Hazewinkel, Michiel, ed. (2001), "Adjoint matrix", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 [Amazon-US | Amazon-UK]
Weisstein, Eric W., "Conjugate Transpose", MathWorld.
Conjugate transpose, PlanetMath.org.

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