Conjugate transpose

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

In mathematics, the conjugate transpose or Hermitian transpose of an m-by-n matrix ${\boldsymbol {A}}$ with complex entries is the n-by-m matrix ${\boldsymbol {A}}^{\mathrm {H} }$ obtained from ${\boldsymbol {A}}$ by taking the transpose and then taking the complex conjugate of each entry. (The complex conjugate of $a+ib$, where $a$ and $b$ are real numbers, is $a-ib$.)

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 subscripts denote 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, bedaggered matrix, 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), universally used in quantum mechanics
• ${\boldsymbol {A}}^{+}$, although this symbol is more commonly used for the Moore–Penrose pseudoinverse

In some contexts (including this article), ${\boldsymbol {A}}^{*}$ 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:

${\boldsymbol {A}}^{\mathrm {T} }={\begin{bmatrix}1&1+i\\-2-i&i\\5&4-2i\end{bmatrix}}$

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 ${\boldsymbol {A}}^{\mathrm {H} }={\boldsymbol {A}}^{-1}$.

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 2×2 real matrices, obeying matrix addition and multiplication:

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

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

An m-by-n matrix of complex numbers could therefore equally well be represented by a 2m-by-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 n-by-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-by-n matrix ${\boldsymbol {A}}$.
• $({\boldsymbol {A}}{\boldsymbol {B}})^{\mathrm {H} }={\boldsymbol {B}}^{\mathrm {H} }{\boldsymbol {A}}^{\mathrm {H} }$ for any m-by-n matrix ${\boldsymbol {A}}$ and any n-by-p matrix ${\boldsymbol {B}}$. Note that the order of the factors is reversed.
• $({\boldsymbol {A}}^{\mathrm {H} })^{\mathrm {H} }=A$ for any m-by-n matrix ${\boldsymbol {A}}$, i.e. Hermitian transposition is an involution.
• If ${\boldsymbol {A}}$ is a square matrix, then $\operatorname {det} ({\boldsymbol {A}}^{\mathrm {H} })={\overline {\operatorname {det} ({\boldsymbol {A}})}}$ where $\operatorname {det} (A)$ denotes the determinant of ${\boldsymbol {A}}$ .
• If ${\boldsymbol {A}}$ is a square matrix, then $\operatorname {tr} ({\boldsymbol {A}}^{\mathrm {H} })={\overline {\operatorname {tr} ({\boldsymbol {A}})}}$ 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 $({\boldsymbol {A}}^{\mathrm {H} })^{-1}=({\boldsymbol {A}}^{-1})^{\mathrm {H} }$.
• The eigenvalues of ${\boldsymbol {A}}^{\mathrm {H} }$ are the complex conjugates of the eigenvalues of ${\boldsymbol {A}}$.
• $\langle {\boldsymbol {A}}x,y\rangle =\langle x,{\boldsymbol {A}}^{\mathrm {H} }y\rangle$ for any m-by-n matrix ${\boldsymbol {A}}$, any vector in $x\in \mathbb {C} ^{n}$ and any vector $y\in \mathbb {C} ^{m}$. Here, $\langle \cdot ,\cdot \rangle$ denotes the standard complex inner product on $\mathbb {C} ^{m}$ and $\mathbb {C} ^{n}$.

Generalizations

The last property given above shows that if one views ${\boldsymbol {A}}$ as a linear transformation from Euclidean 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

• Hermitian adjoint
• Adjugate matrix

External links

• Hazewinkel, Michiel, ed. (2001) [1994], "Adjoint matrix", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
• Weisstein, Eric W. "Conjugate Transpose". MathWorld.
• "Conjugate transpose". PlanetMath.