# Conjugate transpose

In mathematics, the conjugate transpose or Hermitian transpose of an m-by-n matrix ${\boldsymbol }$ with complex entries is the n-by-m matrix ${\boldsymbol }^{\mathrm }$ obtained from ${\boldsymbol }$ by taking the transpose and then taking the complex conjugate of each entry. (The complex conjugate of $a+ib$ , where $a$ and $b$ are real numbers, is $a-ib$ .)

## Definition

The conjugate transpose of an $m\times n$ matrix ${\boldsymbol }$ is formally defined by

$\left({\boldsymbol }^{\mathrm }\right)_={\overline {{\boldsymbol }_}}$ (Eq.1)

where the subscripts denote the $(i,j)$ -th entry, for $1\leq i\leq n$ and $1\leq j\leq m$ , and the overbar denotes a scalar complex conjugate.

This definition can also be written as

${\boldsymbol }^{\mathrm }=\left({\overline {\boldsymbol }}\right)^{\mathsf }={\overline {{\boldsymbol }^{\mathsf }}}$ where ${\boldsymbol }^{\mathsf }$ denotes the transpose and ${\overline {\boldsymbol }}$ denotes the matrix with complex conjugated entries.

Other names for the conjugate transpose of a matrix are Hermitian conjugate, bedaggered matrix, adjoint matrix or transjugate. The conjugate transpose of a matrix ${\boldsymbol }$ can be denoted by any of these symbols:

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

In some contexts, ${\boldsymbol }^{*}$ denotes the matrix with only complex conjugated entries and no transposition.

## Example

Suppose we want to calculate the conjugate transpose of the following matrix ${\boldsymbol }$ .

${\boldsymbol }={\begin1&-2-i&5\\1+i&i&4-2i\end}$ We first transpose the matrix:

${\boldsymbol }^{\mathrm }={\begin1&1+i\\-2-i&i\\5&4-2i\end}$ Then we conjugate every entry of the matrix:

${\boldsymbol }^{\mathrm }={\begin1&1-i\\-2+i&-i\\5&4+2i\end}$ ## Basic remarks

A square matrix ${\boldsymbol }$ with entries $a_$ is called

• Hermitian or self-adjoint if ${\boldsymbol }={\boldsymbol }^{\mathrm }$ ; i.e., $a_={\overline }}$ .
• skew Hermitian or antihermitian if ${\boldsymbol }=-{\boldsymbol }^{\mathrm }$ ; i.e., $a_=-{\overline }}$ .
• normal if ${\boldsymbol }^{\mathrm }{\boldsymbol }={\boldsymbol }{\boldsymbol }^{\mathrm }$ .
• unitary if ${\boldsymbol }^{\mathrm }={\boldsymbol }^{-1}$ .

Even if ${\boldsymbol }$ is not square, the two matrices ${\boldsymbol }^{\mathrm }{\boldsymbol }$ and ${\boldsymbol }{\boldsymbol }^{\mathrm }$ are both Hermitian and in fact positive semi-definite matrices.

The conjugate transpose "adjoint" matrix ${\boldsymbol }^{\mathrm }$ should not be confused with the adjugate, $\operatorname ({\boldsymbol })$ , which is also sometimes called adjoint.

The conjugate transpose of a matrix ${\boldsymbol }$ with real entries reduces to the transpose of ${\boldsymbol }$ , 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:

$a+ib\equiv {\begina&-b\\b&a\end}.$ 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 $\mathbb ^$ ) affected by complex z-multiplication on $\mathbb$ .

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

• $({\boldsymbol }+{\boldsymbol })^{\mathrm }={\boldsymbol }^{\mathrm }+{\boldsymbol }^{\mathrm }$ for any two matrices ${\boldsymbol }$ and ${\boldsymbol }$ of the same dimensions.
• $(z{\boldsymbol })^{\mathrm }={\overline }{\boldsymbol }^{\mathrm }$ for any complex number $z$ and any m-by-n matrix ${\boldsymbol }$ .
• $({\boldsymbol }{\boldsymbol })^{\mathrm }={\boldsymbol }^{\mathrm }{\boldsymbol }^{\mathrm }$ for any m-by-n matrix ${\boldsymbol }$ and any n-by-p matrix ${\boldsymbol }$ . Note that the order of the factors is reversed.
• $({\boldsymbol }^{\mathrm })^{\mathrm }={\boldsymbol }$ for any m-by-n matrix ${\boldsymbol }$ , i.e. Hermitian transposition is an involution.
• If ${\boldsymbol }$ is a square matrix, then $\operatorname ({\boldsymbol }^{\mathrm })={\overline {\operatorname ({\boldsymbol })}}$ where $\operatorname (A)$ denotes the determinant of ${\boldsymbol }$ .
• If ${\boldsymbol }$ is a square matrix, then $\operatorname ({\boldsymbol }^{\mathrm })={\overline {\operatorname ({\boldsymbol })}}$ where $\operatorname (A)$ denotes the trace of ${\boldsymbol }$ .
• ${\boldsymbol }$ is invertible if and only if ${\boldsymbol }^{\mathrm }$ is invertible, and in that case $({\boldsymbol }^{\mathrm })^{-1}=({\boldsymbol }^{-1})^{\mathrm }$ .
• The eigenvalues of ${\boldsymbol }^{\mathrm }$ are the complex conjugates of the eigenvalues of ${\boldsymbol }$ .
• $\langle {\boldsymbol }x,y\rangle _=\langle x,{\boldsymbol }^{\mathrm }y\rangle _$ for any m-by-n matrix ${\boldsymbol }$ , any vector in $x\in \mathbb ^$ and any vector $y\in \mathbb ^$ . Here, $\langle \cdot ,\cdot \rangle _$ denotes the standard complex inner product on $\mathbb ^$ , and similarly for $\langle \cdot ,\cdot \rangle _$ .

## Generalizations

The last property given above shows that if one views ${\boldsymbol }$ as a linear transformation from Hilbert space $\mathbb ^$ to $\mathbb ^,$ then the matrix ${\boldsymbol }^{\mathrm }$ corresponds to the adjoint operator of ${\boldsymbol }$ . The concept of adjoint operators between Hilbert spaces can thus be seen as a generalization of the conjugate transpose of matrices with respect to an orthonormal basis.

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 $V$ .