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## Summary

In mathematics, the conjugate transpose (or Hermitian transpose) of an m-by-n matrix ${\boldsymbol {A}}$ with complex entries is the n-by-m matrix obtained from ${\boldsymbol {A}}$ by taking the transpose and then taking the complex conjugate of each entry (the complex conjugate of $a+ib$ being $a-ib$ , for real numbers $a$ and $b$ ). It is often denoted as ${\boldsymbol {A}}^{\mathrm {H} }$ or ${\boldsymbol {A}}^{*}$ .

For real matrices, the conjugate transpose is just the transpose, ${\boldsymbol {A}}^{\mathrm {H} }={\boldsymbol {A}}^{\mathsf {T}}$ .

## Definition

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

$\left({\boldsymbol {A}}^{\mathrm {H} }\right)_{ij}={\overline {{\boldsymbol {A}}_{ji}}}$

(Eq.1)

where the subscript $ij$  denotes 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

${\boldsymbol {A}}^{\mathrm {H} }=\left({\overline {\boldsymbol {A}}}\right)^{\mathsf {T}}={\overline {{\boldsymbol {A}}^{\mathsf {T}}}}$

where ${\boldsymbol {A}}^{\mathsf {T}}$  denotes the transpose and ${\overline {\boldsymbol {A}}}$  denotes the matrix with complex conjugated entries.

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

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

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

## Example

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

${\boldsymbol {A}}={\begin{bmatrix}1&-2-i&5\\1+i&i&4-2i\end{bmatrix}}$

We first transpose the matrix:

${\boldsymbol {A}}^{\mathsf {T}}={\begin{bmatrix}1&1+i\\-2-i&i\\5&4-2i\end{bmatrix}}$

Then we conjugate every entry of the matrix:

${\boldsymbol {A}}^{\mathrm {H} }={\begin{bmatrix}1&1-i\\-2+i&-i\\5&4+2i\end{bmatrix}}$

## Basic remarks

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

• Hermitian or self-adjoint if ${\boldsymbol {A}}={\boldsymbol {A}}^{\mathrm {H} }$ ; i.e., $a_{ij}={\overline {a_{ji}}}$ .
• Skew Hermitian or antihermitian if ${\boldsymbol {A}}=-{\boldsymbol {A}}^{\mathrm {H} }$ ; i.e., $a_{ij}=-{\overline {a_{ji}}}$ .
• Normal if ${\boldsymbol {A}}^{\mathrm {H} }{\boldsymbol {A}}={\boldsymbol {A}}{\boldsymbol {A}}^{\mathrm {H} }$ .
• Unitary if ${\boldsymbol {A}}^{\mathrm {H} }={\boldsymbol {A}}^{-1}$ , equivalently ${\boldsymbol {A}}{\boldsymbol {A}}^{\mathrm {H} }={\boldsymbol {I}}$ , equivalently ${\boldsymbol {A}}^{\mathrm {H} }{\boldsymbol {A}}={\boldsymbol {I}}$ .

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

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

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

$a+ib\equiv {\begin{bmatrix}a&-b\\b&a\end{bmatrix}}.$

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 $\mathbb {R} ^{2}$ ), affected by complex z-multiplication on $\mathbb {C}$ .

Thus, an m-by-n matrix of complex numbers could be well 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 an n-by-m matrix made up of complex numbers.

## Properties of the conjugate transpose

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

## Generalizations

The last property given above shows that if one views ${\boldsymbol {A}}$  as a linear transformation from Hilbert space $\mathbb {C} ^{n}$  to $\mathbb {C} ^{m},$  then the matrix ${\boldsymbol {A}}^{\mathrm {H} }$  corresponds to the adjoint operator of ${\boldsymbol {A}}$ . 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 $V$ .