In linear algebra, a square matrix with complex entries is said to be skew-Hermitian or anti-Hermitian if its conjugate transpose is the negative of the original matrix. That is, the matrix $A$ is skew-Hermitian if it satisfies the relation

$A{\text{ skew-Hermitian))\quad \iff \quad A^{\mathsf {H))=-A$ where $A^{\textsf {H))$ denotes the conjugate transpose of the matrix $A$ . In component form, this means that

$A{\text{ skew-Hermitian))\quad \iff \quad a_{ij}=-{\overline {a_{ji)))$ for all indices $i$ and $j$ , where $a_{ij)$ is the element in the $i$ -th row and $j$ -th column of $A$ , and the overline denotes complex conjugation.

Skew-Hermitian matrices can be understood as the complex versions of real skew-symmetric matrices, or as the matrix analogue of the purely imaginary numbers. The set of all skew-Hermitian $n\times n$ matrices forms the $u(n)$ Lie algebra, which corresponds to the Lie group U(n). The concept can be generalized to include linear transformations of any complex vector space with a sesquilinear norm.

Note that the adjoint of an operator depends on the scalar product considered on the $n$ dimensional complex or real space $K^{n)$ . If $(\cdot \mid \cdot )$ denotes the scalar product on $K^{n)$ , then saying $A$ is skew-adjoint means that for all $\mathbf {u} ,\mathbf {v} \in K^{n)$ one has $(A\mathbf {u} \mid \mathbf {v} )=-(\mathbf {u} \mid A\mathbf {v} )$ .

Imaginary numbers can be thought of as skew-adjoint (since they are like $1\times 1$ matrices), whereas real numbers correspond to self-adjoint operators.

## Example

For example, the following matrix is skew-Hermitian

$A={\begin{bmatrix}-i&+2+i\\-2+i&0\end{bmatrix))$ because
$-A={\begin{bmatrix}i&-2-i\\2-i&0\end{bmatrix))={\begin{bmatrix}{\overline {-i))&{\overline {-2+i))\\{\overline {2+i))&{\overline {0))\end{bmatrix))={\begin{bmatrix}{\overline {-i))&{\overline {2+i))\\{\overline {-2+i))&{\overline {0))\end{bmatrix))^{\mathsf {T))=A^{\mathsf {H))$ ## Properties

• The eigenvalues of a skew-Hermitian matrix are all purely imaginary (and possibly zero). Furthermore, skew-Hermitian matrices are normal. Hence they are diagonalizable and their eigenvectors for distinct eigenvalues must be orthogonal.
• All entries on the main diagonal of a skew-Hermitian matrix have to be pure imaginary; i.e., on the imaginary axis (the number zero is also considered purely imaginary).
• If $A$ and $B$ are skew-Hermitian, then $aA+bB$ is skew-Hermitian for all real scalars $a$ and $b$ .
• $A$ is skew-Hermitian if and only if $iA$ (or equivalently, $-iA$ ) is Hermitian.
• $A$ is skew-Hermitian if and only if the real part $\Re {(A))$ is skew-symmetric and the imaginary part $\Im {(A))$ is symmetric.
• If $A$ is skew-Hermitian, then $A^{k)$ is Hermitian if $k$ is an even integer and skew-Hermitian if $k$ is an odd integer.
• $A$ is skew-Hermitian if and only if $\mathbf {x} ^{\mathsf {H))A\mathbf {y} =-\mathbf {y} ^{\mathsf {H))A\mathbf {x}$ for all vectors $\mathbf {x} ,\mathbf {y}$ .
• If $A$ is skew-Hermitian, then the matrix exponential $e^{A)$ is unitary.
• The space of skew-Hermitian matrices forms the Lie algebra $u(n)$ of the Lie group $U(n)$ .

## Decomposition into Hermitian and skew-Hermitian

• The sum of a square matrix and its conjugate transpose $\left(A+A^{\mathsf {H))\right)$ is Hermitian.
• The difference of a square matrix and its conjugate transpose $\left(A-A^{\mathsf {H))\right)$ is skew-Hermitian. This implies that the commutator of two Hermitian matrices is skew-Hermitian.
• An arbitrary square matrix $C$ can be written as the sum of a Hermitian matrix $A$ and a skew-Hermitian matrix $B$ :
$C=A+B\quad {\mbox{with))\quad A={\frac {1}{2))\left(C+C^{\mathsf {H))\right)\quad {\mbox{and))\quad B={\frac {1}{2))\left(C-C^{\mathsf {H))\right)$ 