On the product of two skew-Hamiltonian matrices or two skew-symmetric matrices

We show that the product $C$ of two skew-Hamiltonian matrices obeys the Stenzel conditions. If at least one of the factors is nonsingular, then the Stenzel conditions amount to the requirement that every elementary divisor for a nonzero eigenvalue of $C$ occurs an even number of times. The same properties are valid for the product of two skew-pseudosymmetric matrices. We observe that the method proposed by Van Loan for computing the eigenvalues of real Hamiltonian and skew-Hamiltonian matrices can be extended to complex skew-Hamiltonian matrices. Finally, we show that the computation of the eigenvalues of a product of two skew-symmetric matrices can be reduced to computing the eigenvalues of a similar skew-Hamiltonian matrix. Bibl. – 8 titles.