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 A^* is also its negative. That is, if it satisfies the relation:

A^* = −A

or in component form, if A = (a_i,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>

All entries on the main diagonal of a skew-Hermitian matrix have to be pure imaginary.

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