Generic canonical forms for perplectic and symplectic normal matrices

TL;DR

This paper develops sparse canonical forms for certain classes of normal matrices under specific similarity transformations, showing these forms are generic for most such matrices in complex spaces.

Contribution

It introduces new canonical forms for nondefective $J_{2n}$-normal and $R_n$-normal matrices under $J_{2n}$-unitary and $R_n$-unitary similarity, respectively, demonstrating their generic existence.

Findings

Canonical forms exist for an open and dense subset of $J_{2n}$-normal matrices.

Canonical forms exist for an open and dense subset of $R_n$-normal matrices.

These forms are topologically 'generic' for the respective classes of 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.