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Conjugate transpose

## Summary

In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an ${\displaystyle m\times n}$ complex matrix ${\displaystyle \mathbf {A} }$ is an ${\displaystyle n\times m}$ matrix obtained by transposing ${\displaystyle \mathbf {A} }$ and applying complex conjugation to each entry (the complex conjugate of ${\displaystyle a+ib}$ being ${\displaystyle a-ib}$, for real numbers ${\displaystyle a}$ and ${\displaystyle b}$). There are several notations, such as ${\displaystyle \mathbf {A} ^{\mathrm {H} }}$ or ${\displaystyle \mathbf {A} ^{*}}$,[1] ${\displaystyle \mathbf {A} '}$,[2] or (often in physics) ${\displaystyle \mathbf {A} ^{\dagger }}$.

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

## Definition

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

${\displaystyle \left(\mathbf {A} ^{\mathrm {H} }\right)_{ij}={\overline {\mathbf {A} _{ji}}}}$
(Eq.1)

where the subscript ${\displaystyle ij}$  denotes 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

${\displaystyle \mathbf {A} ^{\mathrm {H} }=\left({\overline {\mathbf {A} }}\right)^{\operatorname {T} }={\overline {\mathbf {A} ^{\operatorname {T} }}}}$

where ${\displaystyle \mathbf {A} ^{\operatorname {T} }}$  denotes the transpose and ${\displaystyle {\overline {\mathbf {A} }}}$  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 ${\displaystyle \mathbf {A} }$  can be denoted by any of these symbols:

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

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

## Example

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

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

We first transpose the matrix:

${\displaystyle \mathbf {A} ^{\operatorname {T} }={\begin{bmatrix}1&1+i\\-2-i&i\\5&4-2i\end{bmatrix}}}$

Then we conjugate every entry of the matrix:

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

## Basic remarks

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

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

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

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

The conjugate transpose of a matrix ${\displaystyle \mathbf {A} }$  with real entries reduces to the transpose of ${\displaystyle \mathbf {A} }$ , as the conjugate of a real number is the number itself.

The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by ${\displaystyle 2\times 2}$  real matrices, obeying matrix addition and multiplication:

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

That is, denoting each complex number ${\displaystyle z}$  by the real ${\displaystyle 2\times 2}$  matrix of the linear transformation on the Argand diagram (viewed as the real vector space ${\displaystyle \mathbb {R} ^{2}}$ ), affected by complex ${\displaystyle z}$ -multiplication on ${\displaystyle \mathbb {C} }$ .

Thus, an ${\displaystyle m\times n}$  matrix of complex numbers could be well represented by a ${\displaystyle 2m\times 2n}$  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 ${\displaystyle n\times m}$  matrix made up of complex numbers.

For an explanation of the notation used here, we begin by representing complex numbers ${\displaystyle e^{i\theta }}$  as the rotation matrix, that is,

${\displaystyle e^{i\theta }={\begin{pmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{pmatrix}}=\cos \theta {\begin{pmatrix}1&0\\0&1\end{pmatrix}}+\sin \theta {\begin{pmatrix}0&1\\-1&0\end{pmatrix}}.}$

Since ${\displaystyle e^{i\theta }=\cos \theta +i\sin \theta }$  we are led to the matrix representations of the unit numbers as

${\displaystyle 1={\begin{pmatrix}1&0\\0&1\end{pmatrix}},\quad i={\begin{pmatrix}0&1\\-1&0\end{pmatrix}}.}$  A general complex number ${\displaystyle z=x+iy}$  is then represented as

${\displaystyle z={\begin{pmatrix}x&y\\-y&x\end{pmatrix}}.}$  The complex conjugate operation, where i→−i, is seen to be just the matrix transpose.

## Properties of the conjugate transpose

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

## Generalizations

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