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 ${\boldsymbol {A}}$ is an $n\times m$ matrix obtained by transposing ${\boldsymbol {A}}$ and applying complex conjugate on 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 ${\boldsymbol {A}}^{*}$ or ${\boldsymbol {A}}'$, and very commonly in physics as ${\boldsymbol {A}}^{\dagger }$.

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 $2\times 2$ 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.

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 $m\times n$ matrix ${\boldsymbol {A}}$.
• $({\boldsymbol {A}}{\boldsymbol {B}})^{\mathrm {H} }={\boldsymbol {B}}^{\mathrm {H} }{\boldsymbol {A}}^{\mathrm {H} }$ for any $m\times n$ matrix ${\boldsymbol {A}}$ and any $n\times p$ matrix ${\boldsymbol {B}}$. Note that the order of the factors is reversed.
• $\left({\boldsymbol {A}}^{\mathrm {H} }\right)^{\mathrm {H} }={\boldsymbol {A}}$ for any $m\times n$ 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 $m\times n$ 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 $V$.

See also

• Complex dot product
• Hermitian adjoint
• Adjugate matrix

External links

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