Skew-Hermitian matrix

Skew-Hermitian matrix
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In linear algebra, a square matrix (or more generally, a linear transformation from a complex vector space with a sesquilinear norm to itself) A is said to be skew-Hermitian or antihermitian if its conjugate transpose A^* is also its negative. That is, if it satisfies the relation:

A^* = −A

or in component form, if A = (a_i,j):

$a_{i,j} = -\overline{a_{j,i}}$

for all i and j.

Examples

For example, the following matrix is skew-Hermitian:

$\begin{pmatrix}i & 2 + i \\ -2 + i & 3i \end{pmatrix}$

The eigenvalues of a skew-Hermitian matrix are all purely imaginary. 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, ie. on the imaginary axis. This definition includes the number 0i.
If A is skew-Hermitian, then iA is Hermitian
If A, B are skew-Hermitian, then aA + bB is skew-Hermitian for all real scalars a, b.
If A is skew-Hermitian, then A^2k is Hermitian for all positive integers k.
If A is skew-Hermitian, then A raised to an odd power is skew-Hermitian.
If A is skew-Hermitian, then e^A is unitary.
The difference of a matrix and its conjugate transpose ( $C - C *$ ) 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:

$C = A+B \quad\mbox{with}\quad A = \frac{1}{2}(C + C^*) \quad\mbox{and}\quad B = \frac{1}{2}(C - C^*).$

The space of skew-Hermitian matrices forms the Lie algebra u(n) of the Lie group U(n).