# Skew-Hermitian

An ${\displaystyle n}$ by ${\displaystyle n}$ complex or real matrix ${\displaystyle A=(a_{i,j})_{1\leq i,j\leq n}}$ 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 ${\displaystyle n}$ dimensional space ${\displaystyle K^{n}}$, if its adjoint is the negative of itself: :${\displaystyle A^{*}=-A}$.
Note that the adjoint of an operator depends on the scalar product considered on the ${\displaystyle n}$ dimensional complex or real space ${\displaystyle K^{n}}$. If ${\displaystyle (\cdot |\cdot )}$ denotes the scalar product on ${\displaystyle K^{n}}$, then saying ${\displaystyle A}$ is skew-adjoint means that for all ${\displaystyle u,v\in K^{n}}$ one has ${\displaystyle (Au|v)=-(u|Av)\,.}$
In the particular case of the canonical scalar products on ${\displaystyle K^{n}}$, the matrix of a skew-adjoint operator satisfies ${\displaystyle a_{ij}=-{\overline {a}}_{ji}}$ for all ${\displaystyle 1\leq i,j\leq n}$.