Simple forms for perplectic and symplectic normal matrices

Abstract

Let B=J2n or B=Rn for the matrices given byJ2n=[In−In]∈M2n(C)orRn=[1⋰1]∈Mn(C). A matrix A is called B-normal if AA⋆=A⋆A holds for A and its adjoint matrix A⋆:=B−1AHB. In addition, a matrix Q is called B-unitary, if QHBQ=B. We develop sparse simple forms for nondefective (i.e. diagonalizable) J2n/Rn-normal matrices under J2n/Rn-unitary similarity transformations. For both cases we show that these forms exist for an open and dense subset of J2n/Rn-normal matrices. This implies that these forms can be seen as topologically ‘generic’ since they exist for all J2n/Rn-normal matrices except a nowhere dense subset.