# Conjugate transpose

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

## Definition

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

${\displaystyle \left({\boldsymbol }^{\mathrm }\right)_={\overline {{\boldsymbol }_}}}$

(Eq.1)

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

This definition can also be written as

${\displaystyle {\boldsymbol }^{\mathrm }=\left({\overline {\boldsymbol }}\right)^{\mathsf }={\overline {{\boldsymbol }^{\mathsf }}}}$

where ${\displaystyle {\boldsymbol }^{\mathsf }}$ denotes the transpose and ${\displaystyle {\overline {\boldsymbol }}}$ 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 ${\displaystyle {\boldsymbol }}$ can be denoted by any of these symbols:

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

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

## Example

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

${\displaystyle {\boldsymbol }={\begin1&-2-i&5\\1+i&i&4-2i\end}}$

We first transpose the matrix:

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

Then we conjugate every entry of the matrix:

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

## Basic remarks

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

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

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

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

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

${\displaystyle a+ib\equiv {\begina&-b\\b&a\end}.}$

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 ${\displaystyle \mathbb ^}$) affected by complex z-multiplication on ${\displaystyle \mathbb }$.

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

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

## Generalizations

The last property given above shows that if one views ${\displaystyle {\boldsymbol }}$ as a linear transformation from Hilbert space ${\displaystyle \mathbb ^}$ to ${\displaystyle \mathbb ^,}$ then the matrix ${\displaystyle {\boldsymbol }^{\mathrm }}$ corresponds to the adjoint operator of ${\displaystyle {\boldsymbol }}$. 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 ${\displaystyle A}$ is a linear map from a complex vector space ${\displaystyle V}$ to another, ${\displaystyle W}$, then the complex conjugate linear map as well as the transposed linear map are defined, and we may thus take the conjugate transpose of ${\displaystyle A}$ to be the complex conjugate of the transpose of ${\displaystyle A}$. It maps the conjugate dual of ${\displaystyle W}$ to the conjugate dual of ${\displaystyle V}$.