Antilinear Operators and Their Matrix Presentation under Two Special Scalar Products

Let square matrix $J=diag (\pm1)$. We assume that $\mathbb{C}$$^{n}$ is equipped with one of the following unitary scalar products $(x,y)_{H}=(Jx,y)_{U}$ or $(x,y)_{K}=(Jx,y)_{E}$ where $(x,y)_{U}=y^{\ast}x=x_{1}\overline{y_{1}}+x_{2}\overline{y_{2}}+...+x_{n}\overline{y_{n}}$ and $(x,y)_{E}=y^{T}x=x_{1}y_{1}+x_{2}y_{2}+...+x_{n}y_{n}.$ Our purpose in this paper is to find out which classes of special matrices are distinguished in the case where $n$-by-$n$ matrices are interpreted as antilinear operators in the space $\mathbb{C}$$^{n}$ equipped with scalar product $(x,y)_{H}$ or $(x,y)_{K}$.