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Conjugate transpose

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From Wikipedia, the free encyclopedia

Complex matrix A* obtained from a matrix A by transposing it and conjugating each entry

"Adjoint matrix" redirects here. For the transpose of cofactor, seeAdjugate matrix.

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

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

Definition

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

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

Eq.1

where the subscript $i j {\displaystyle ij}$ denotes the $(i,j)$ -th entry (matrix element), 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

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

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

Other names for the conjugate transpose of a matrix areHermitian transpose,Hermitian conjugate,adjoint matrix ortransjugate. The conjugate transpose of a matrix $\mathbf {A}$ can be denoted by any of these symbols:

$\mathbf {A} ^{*}$ , commonly used inlinear algebra
$\mathbf {A} ^{\mathrm {H} }$ , commonly used in linear algebra
$\mathbf {A} ^{\dagger }$ (sometimes pronounced asAdagger), commonly used inquantum mechanics
$\mathbf {A} ^{+}$ , although this symbol is more commonly used for theMoore–Penrose pseudoinverse

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

Example

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

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

We first transpose the matrix:

\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:

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

Basic remarks

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

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

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

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

The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by $2\times 2$ real matrices, obeyingmatrix addition and multiplication: $a+ib\equiv {\begin{bmatrix}a&-b\\b&a\end{bmatrix}}.$

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

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

For an explanation of the notation used here, we begin by representing complex numbers $e^{i\theta }$ as therotation matrix, that is, $e^{i\theta }={\begin{pmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{pmatrix}}=\cos \theta {\begin{pmatrix}1&0\\0&1\end{pmatrix}}+\sin \theta {\begin{pmatrix}0&-1\\1&0\end{pmatrix}}.$ Since $e^{i\theta }=\cos \theta +i\sin \theta$ , we are led to the matrix representations of the unit numbers as $1={\begin{pmatrix}1&0\\0&1\end{pmatrix}},\quad i={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}.$

A general complex number $z=x+iy$ is then represented as $z={\begin{pmatrix}x&-y\\y&x\end{pmatrix}}.$ Thecomplex conjugate operation (that sends $a+bi$ to $a-bi$ for real $a, b {\displaystyle a,b}$ ) is encoded as the matrix transpose.^[3]

Properties

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

Generalizations

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The last property given above shows that if one views $\mathbf {A}$ as alinear transformation fromHilbert space $\mathbb {C} ^{n}$ to $\mathbb {C} ^{m},$ then the matrix $\mathbf {A} ^{\mathrm {H} }$ corresponds to theadjoint operator of $\mathbf {A}$ . 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 $A {\displaystyle A}$ is a linear map from a complexvector space $V {\displaystyle V}$ to another, $W {\displaystyle W}$ , then thecomplex conjugate linear map as well as thetransposed linear map are defined, and we may thus take the conjugate transpose of $A {\displaystyle A}$ to be the complex conjugate of the transpose of $A {\displaystyle A}$ . It maps the conjugatedual of $W {\displaystyle W}$ to the conjugate dual of $V {\displaystyle V}$ .

References

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^^a ^bWeisstein, Eric W."Conjugate Transpose".mathworld.wolfram.com. Retrieved2020-09-08.
^H. W. Turnbull, A. C. Aitken,"An Introduction to the Theory of Canonical Matrices,"1932.
^Chasnov, Jeffrey R. (4 February 2022). "1.6: Matrix Representation of Complex Numbers".Applied Linear Algebra and Differential Equations. LibreTexts.