# Conjugate transpose

In mathematics, the conjugate transpose 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. Formally

$\displaystyle (A^*)[i,j] = \overline{A[j,i]}$

for 1 ≤ in and 1 ≤ jm.

This definition can also be written as

$\displaystyle A^* \equiv {\overline A}^{T}$

where $\displaystyle A^T \,\!$ denotes the transpose and $\displaystyle \overline A \,\!$ denotes the matrix with complex conjugated entries.

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

• $\displaystyle A^* \,\!$
• $\displaystyle A^H \,\!$ , commonly used in linear algebra
• $\displaystyle A^\dagger \,\!$ , universally used in quantum field theory

In some contexts $\displaystyle A^* \,\!$ can be used to denote the complex conjugate so care must be taken not to confuse notations.

## Example

If

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

then

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

## 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 is called

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

## Properties of the conjugate transpose

• (A + B)* = A* + B* for any two matrices A and B of the same format.
• (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 trace (A*) = (trace 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 ordinary Euclidean inner product (or dot product) on Cm and Cn.

## Generalizations

The last property given above shows that if one views A as a linear map 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.