Conjugate transpose

Example

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

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

We first transpose the matrix:

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

Then we conjugate every entry of the matrix:

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

Basic remarks

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

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, ${\displaystyle \mathrm{adj}(\mathbf{A})}$, which is also sometimes called adjoint.

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

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, $${\displaystyle e^{i\theta }=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)=\cos \theta \left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)+\sin \theta \left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right).}$$ Since ${\displaystyle e^{i\theta }=\cos \theta +i\sin \theta }$, we are led to the matrix representations of the unit numbers as $${\displaystyle 1=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),i=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right).}$$

A general complex number ${\displaystyle z=x+iy}$ is then represented as ${\displaystyle z=\left(\begin{array}{cc}x& -y\\ y& x\end{array}\right).}$ The complex conjugate operation (that sends ${\displaystyle a+bi}$ to ${\displaystyle a-bi}$ for real ${\displaystyle a,b}$) is encoded as the matrix transpose.

Properties

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

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 ${\displaystyle V}$.