Skew-Hermitian

From Wikipedia, the free encyclopedia
Jump to: navigation, search

An n by n complex or real matrix A = (a_{i,j})_{1 \leq i, j \leq n} is said to be anti-Hermitian, skew-Hermitian, or said to represent a skew-adjoint operator, or to be a skew-adjoint matrix, on the complex or real n dimensional space K^n, if its adjoint is the negative of itself: :A^*=-A.

Note that the adjoint of an operator depends on the scalar product considered on the n dimensional complex or real space K^n. If (\cdot|\cdot) denotes the scalar product on  K^n, then saying  A is skew-adjoint means that for all u,v \in K^n one has  (Au|v) = - (u|Av) \, .

In the particular case of the canonical scalar products on K^n, the matrix of a skew-adjoint operator satisfies a_{ij} = - {\overline a}_{ji} for all 1 \leq i,j \leq n.

Imaginary numbers can be thought of as skew-adjoint (since they are like 1-by-1 matrices), whereas real numbers correspond to self-adjoint operators.

See also[edit]