In linear algebra, a square matrix with complex entries is said to be skew-Hermitian or anti-Hermitian if its conjugate transpose is the negative of the original matrix. That is, the matrix A is skew-Hermitian if it satisfies the relation where denotes the conjugate transpose of the matrix A. In component form, this means that for all indices i and j, where a_{ij} is the element in the i-th row and j-th column of A, and the overline denotes complex conjugation. Skew-Hermitian matrices can be understood as the complex versions of real skew-symmetric matrices, or as the matrix analogue of the purely imaginary numbers. The set of all skew-Hermitian n \times n matrices forms the u(n) Lie algebra, which corresponds to the Lie group U( n ). The concept can be generalized to include linear transformations of any complex vector space with a sesquilinear norm. Note that the adjoint of an operator depends on the scalar product considered on the n dimensional complex or real space K^n. If denotes the scalar product on K^n, then saying A is skew-adjoint means that for all one has. Imaginary numbers can be thought of as skew-adjoint (since they are like 1 \times 1 matrices), whereas real numbers correspond to self-adjoint operators.

Example

For example, the following matrix is skew-Hermitian because

Properties

Decomposition into Hermitian and skew-Hermitian