Definitions

# Conjugate transpose

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
$\left(A^*\right)_\left\{ij\right\} = overline\left\{A_\left\{ji\right\}\right\}$
where the subscripts denote the i,j-th entry, for 1 ≤ in and 1 ≤ jm, 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

$A^* = \left(overline\left\{A\right\}\right)^mathrm\left\{T\right\} = overline\left\{A^mathrm\left\{T\right\}\right\}$
where $A^mathrm\left\{T\right\} ,!$ 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 $A^mathrm\left\{H\right\} ,!$, 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.

## Example

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

## Basic remarks

If the entries of A are real, then A* coincides with the transpose AT 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 $a_\left\{ij\right\}$ is called

• Hermitian or self-adjoint if A = A*, i.e., $a_\left\{ij\right\}=a_\left\{ji\right\}^*$ ;
• skew Hermitian or antihermitian if A = −A*, i.e., $a_\left\{ij\right\}=-a_\left\{ji\right\}^\left\{*\right\}$ ;
• 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").

## 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:

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

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.

## 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 Cm. Here <·,·> denotes the standard complex inner product on Cm and Cn.

## Generalizations

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