Conjugate transpose

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

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

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

Definition

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 $ij$ 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

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

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

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

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

Example

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

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

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

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

• Hermitian or self-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 fact positive semi-definite matrices.

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

The conjugate transpose of a matrix $\mathbf {A}$ with real entries reduces to the transpose of $\mathbf {A}$, 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 $2\times 2$ 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 $2\times 2$ 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 $m\times n$ 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 the rotation 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

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

Properties of the conjugate transpose

• $(\mathbf {A} +{\boldsymbol {B}})^{\mathrm {H} }=\mathbf {A} ^{\mathrm {H} }+{\boldsymbol {B}}^{\mathrm {H} }$ 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 $z$ 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.
• $\left(\mathbf {A} ^{\mathrm {H} }\right)^{\mathrm {H} }=\mathbf {A}$ for any $m\times n$ matrix $\mathbf {A}$, i.e. Hermitian transposition is an involution.
• 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 the determinant 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 the trace of $\mathbf {A}$.
• $\mathbf {A}$ is invertible if and only if $\mathbf {A} ^{\mathrm {H} }$ is invertible, and in that case $\left(\mathbf {A} ^{\mathrm {H} }\right)^{-1}=\left(\mathbf {A} ^{-1}\right)^{\mathrm {H} }$.
• The eigenvalues of $\mathbf {A} ^{\mathrm {H} }$ are the complex conjugates of the eigenvalues 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 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 $\mathbf {A}$ as a linear transformation from Hilbert space $\mathbb {C} ^{n}$ to $\mathbb {C} ^{m},$ then the matrix $\mathbf {A} ^{\mathrm {H} }$ corresponds to the adjoint 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 $A$ 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 $V$.

See also

• Complex dot product
• Hermitian adjoint
• Adjugate matrix

External links

• "Adjoint matrix", Encyclopedia of Mathematics, EMS Press, 2001 [1994]