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