Generic canonical forms for perplectic and symplectic normal matrices

06/27/2020

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by   Ralph John de la Cruz, et al.

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Let B be some invertible Hermitian or skew-Hermitian matrix. A matrix A is called B-normal if AA^⋆ = A^⋆ A holds for A and its adjoint matrix A^⋆ := B^-1A^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 = < b m a t r i x > ∈ M_2n(ℂ) and R_n is the n × 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.