Skew-Hermitian matrix
From Wikipedia, the free encyclopedia
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 = (ai,j):
for all i and j.
[edit] Examples
For example, the following matrix is skew-Hermitian:
[edit] Properties
- 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.
- 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 A2k 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 eA 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:
- The space of skew-Hermitian matrices forms the Lie algebra u(n) of the Lie group U(n).
