Skew-Hermitian

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

• Skew-Hermitian matrix