Skew-Hermitian matrix

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 is its negative.^[1] That is, if it satisfies the relation:

$A^\dagger = -A\;$

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

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

for all i and j.

Skew-Hermitian matrices can be understood as the matrix analogue of the purely imaginary numbers.^[2]

[edit] Example

For example, the following matrix is skew-Hermitian:

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

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.^[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 A, B are skew-Hermitian, then aA + bB is skew-Hermitian for all real scalars a and b.^[5]
If A is skew-Hermitian, then iA is Hermitian.^[5]
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.
An arbitrary (square) matrix C can uniquely be written as the sum of a Hermitian matrix A and a skew-Hermitian matrix B:^[2]

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

If A is skew-Hermitian, then e^A is unitary.
The space of skew-Hermitian matrices forms the Lie algebra u(n) of the Lie group U(n).