Generic canonical forms for perplectic and symplectic normal matrices

Abstract

Let $B$ be some invertible Hermitian or skew-Hermitian matrix. A matrix $A$ is called $B$-normal if $AA^\star = A^\star A$ holds for $A$ and its adjoint matrix $A^\star := B^{-1}A^HB$. In addition, a matrix $Q$ is called $B$-unitary, if $Q^HBQ = B$. We develop sparse canonical forms for nondefective (i.e. diagonalizable) $J_{2n}$-normal matrices and $R_n$-normal matrices under $J_{2n}$-unitary ($R_n$-unitary, respectively) similarity transformations where $$J_{2n} = \begin{bmatrix} & I_n \\ - I_n & \end{bmatrix} \in M_{2n}(\mathbb{C})$$ and $R_n$ is the $n \times n$ sip matrix with ones on its anti-diagonal and zeros elsewhere. For both cases we show that these forms exist for an open and dense subset of $J_{2n}/R_n$-normal matrices. This implies that these forms can be seen as topologically 'generic' for $J_{2n}/R_n$-normal matrices since they exist for all such matrices except a nowhere dense subset.