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 = (ai,j):
- <math>a_{i,j} = -\overline{a_{j,i}}<math>
for all i and j.
Examples
For example, the following matrix is skew-Hermitian:
- <math>\begin{pmatrix}i & 2 + i \\ -2 + i & 3i \end{pmatrix}<math>
Properties
- All entries on the main diagonal of a skew-Hermitian matrix have to be pure imaginary, ie. on the imaginary axis. Hence the same is true for the eigenvalues of a skew-Hermitian matrix.
- 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.
- All skew-Hermitian matrices are normal.
- If A is skew-Hermitian, then A2 is Hermitian.
- If A is skew-Hermitian, then A raised to an odd power is skew-Hermitian.