Skew-Hermitian matrix

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 ${\displaystyle A}$ and ${\displaystyle B}$ are skew-Hermitian, then ⁠${\displaystyle aA+bB}$⁠ is skew-Hermitian for all real scalars ${\displaystyle a}$ and ${\displaystyle b}$.
• ${\displaystyle A}$ is skew-Hermitian if and only if ${\displaystyle iA}$ (or equivalently, ${\displaystyle -iA}$) is Hermitian.
• ${\displaystyle A}$ is skew-Hermitian if and only if the real part ${\displaystyle \mathrm{\Re }(A)}$ is skew-symmetric and the imaginary part ${\displaystyle \mathrm{\Im }(A)}$ is symmetric.
• If ${\displaystyle A}$ is skew-Hermitian, then ${\displaystyle A^k}$ is Hermitian if ${\displaystyle k}$ is an even integer and skew-Hermitian if ${\displaystyle k}$ is an odd integer.
• ${\displaystyle A}$ is skew-Hermitian if and only if ${\displaystyle {\mathbf{x}}^{\mathsf{H}}A\mathbf{y}=-\overline{{\mathbf{y}}^{\mathsf{H}}A\mathbf{x}}}$ for all vectors ${\displaystyle \mathbf{x},\mathbf{y}}$.
• If ${\displaystyle A}$ is skew-Hermitian, then the matrix exponential ${\displaystyle e^A}$ is unitary.
• The space of skew-Hermitian matrices forms the Lie algebra ${\displaystyle u(n)}$ of the Lie group ${\displaystyle U(n)}$.

Decomposition into Hermitian and skew-Hermitian

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