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)}}  isskew-symmetric and the imaginary part ${\displaystyle \Im {($A)}}  issymmetric.
• 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 ${\displaystyle e^{A}}$ isunitary.
• 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

• Horn, Roger A.; Johnson, Charles R. (1985), Matrix Analysis, Cambridge University Press, ISBN 978-0-521-38632-6.
• Meyer, Carl D. (2000), Matrix Analysis and Applied Linear Algebra, SIAM, ISBN 978-0-89871-454-8.