Skew-Hermitian matrix

For example, the following matrix is skew-Hermitian $${\displaystyle A={\begin{bmatrix}-i&+2+i\\-2+i&0\end{bmatrix}}}$$ because $${\displaystyle -A={\begin{bmatrix}i&-2-i\\2-i&0\end{bmatrix}}={\begin{bmatrix}{\overline {-i}}&{\overline {-2+i}}\\{\overline {2+i}}&{\overline {0}}\end{bmatrix}}={\begin{bmatrix}{\overline {-i}}&{\overline {2+i}}\\{\overline {-2+i}}&{\overline {0}}\end{bmatrix}}^{\mathsf {T}}=A^{\mathsf {H}}}$$

• 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.^[3]
• 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).^[4]
• If ${\displaystyle A}$ and ${\displaystyle B}$ are skew-Hermitian, then ⁠${\displaystyle aA+bB}$⁠ is skew-Hermitian for all real scalars ${\displaystyle a}$ and ${\displaystyle b}$.^[5]
• ${\displaystyle A}$ is skew-Hermitian if and only if ${\displaystyle iA}$ (or equivalently, ${\displaystyle -iA}$) is Hermitian.^[5]
• ${\displaystyle A}$ is skew-Hermitian if and only if the real part ${\displaystyle \Re {(A)}}$ is skew-symmetric and the imaginary part ${\displaystyle \Im {(A)}}$ 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 ${\displaystyle U(n)}$.

• 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 ${\displaystyle B}$: $${\displaystyle C=A+B\quad {\mbox{with}}\quad A={\frac {1}{2}}\left(C+C^{\mathsf {H}}\right)\quad {\mbox{and}}\quad B={\frac {1}{2}}\left(C-C^{\mathsf {H}}\right)}$$

• Bivector (complex)
• Hermitian matrix
• Normal matrix
• Skew-symmetric matrix
• Unitary matrix