Conjugate transpose

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.

^[3]

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 ${\displaystyle V}$.

See also

• Complex dot product
• Hermitian adjoint
• Adjugate matrix

References

External links

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