Conjugate transpose

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

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 ${\mathbf{𝐀}}^{*}$, ${\mathbf{𝐀}}^{\prime }$, or (often in physics) ${\mathbf{𝐀}}^{†}$.

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

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 $1\le i\le n$ and $1\le j\le m$, and the overbar denotes a scalar complex conjugate.

This definition can also be written as

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

where ${\mathbf{𝐀}}^{\mathrm{T}}$ 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
• ${\mathbf{𝐀}}^{†}$ (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 $\mathbf{𝐀}$.

$\mathbf{𝐀}=\left[\begin{array}{ccc}1& -2-i& 5\\ 1+i& i& 4-2i\end{array}\right]$

We first transpose the matrix:

${\mathbf{𝐀}}^{\mathrm{T}}=\left[\begin{array}{cc}1& 1+i\\ -2-i& i\\ 5& 4-2i\end{array}\right]$

Then we conjugate every entry of the matrix:

${\mathbf{𝐀}}^{\mathrm{H}}=\left[\begin{array}{cc}1& 1-i\\ -2+i& -i\\ 5& 4+2i\end{array}\right]$

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 $\mathbf{𝐀}{\mathbf{𝐀}}^{\mathrm{H}}=\mathit{I}$, equivalently ${\mathbf{𝐀}}^{\mathrm{H}}\mathbf{𝐀}=\mathit{I}$.

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, $\mathrm{adj}(\mathbf{𝐀})$, 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.

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:

$a+ib\equiv \left[\begin{array}{cc}a& -b\\ b& a\end{array}\right].$

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

• $(\mathbf{𝐀}+\mathit{B}{)}^{\mathrm{H}}={\mathbf{𝐀}}^{\mathrm{H}}+{\mathit{B}}^{\mathrm{H}}$ for any two matrices $\mathbf{𝐀}$ and $\mathit{B}$ of the same dimensions.
• $(z\mathbf{𝐀}{)}^{\mathrm{H}}=\bar{z}{\mathbf{𝐀}}^{\mathrm{H}}$ for any complex number $z$ and any $m\times n$ matrix $\mathbf{𝐀}$.
• $(\mathbf{𝐀}\mathit{B}{)}^{\mathrm{H}}={\mathit{B}}^{\mathrm{H}}{\mathbf{𝐀}}^{\mathrm{H}}$ for any $m\times n$ matrix $\mathbf{𝐀}$ and any $n\times p$ matrix $\mathit{B}$. 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 $\det (A)$ denotes the determinant of $\mathbf{𝐀}$ .
• If $\mathbf{𝐀}$ is a square matrix, then $\mathrm{tr}\left({\mathbf{𝐀}}^{\mathrm{H}}\right)=\overline{\mathrm{tr}(\mathbf{𝐀})}$ where $\mathrm{tr}(A)$ 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 $\mathbf{𝐀}$.
• ${⟨\mathbf{𝐀}x,y⟩}_m={⟨x,{\mathbf{𝐀}}^{\mathrm{H}}y⟩}_n$ 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, $⟨\cdot ,\cdot {⟩}_m$ denotes the standard complex inner product on ${\mathbb{ℂ}}^m$, and similarly for $⟨\cdot ,\cdot {⟩}_n$.

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

• Hazewinkel, Michiel, ed. (2001), "Adjoint matrix", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/a010850