# 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

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (A^*)[i,j] = \overline{A[j,i]}}**

for 1 ≤ *i* ≤ *n* and 1 ≤ *j* ≤ *m*.

This definition can also be written as

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^* \equiv {\overline A}^{T}}**

where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^T \,\!}**
denotes the transpose and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^* \,\!}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^H \,\!}**, commonly used in linear algebra**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^\dagger \,\!}**, universally used in quantum field theory

In some contexts **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^* \,\!}**
can be used to denote the complex conjugate so care must be taken not to confuse notations.

## Contents

## Example

If

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=\begin{bmatrix}3+i&2\\ 2-2i&i\end{bmatrix}}**

then

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 *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* is called

- Hermitian or self-adjoint if
*A*=*A*^{*}; - skew Hermitian if
*A*= -*A*^{*}; - 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.

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

## 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**C**^{n}and any vector*y*in**C**^{m}. Here <.,.> denotes the ordinary Euclidean inner product (or dot product) on**C**^{m}and**C**^{n}.

## Generalizations

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

## See also

## External links

- Conjugate transpose on Mathworld, Wolfram research
- Conjugate transpose on PlanetMath.