Conjugate transpose

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"Adjoint matrix" redirects here. An adjugate matrix is sometimes called a "classical adjoint matrix".

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

$(A^*)_{i,j} = \overline{A_{j,i}}$

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

This definition can also be written as

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

where $A^\mathrm{T} \,\!$ denotes the transpose and $\overline A \,\!$ 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:

$A^* \,\!$ 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.

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

[edit] Example

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

then

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

[edit] Basic remarks

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 $a i j$ 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").

[edit] Motivation

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.

[edit] Properties of the conjugate transpose

(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 Cⁿ and any vector y in C^m. Here <·,·> denotes the standard complex inner product on C^m and Cⁿ.

[edit] Generalizations

The last property given above shows that if one views A as a linear transformation from the Euclidean Hilbert space Cⁿ 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.