Several observations on symplectic, Hamiltonian, and skew-Hamiltonian matrices (Q1779250)

There is a classical theorem relating strict equivalence and \(T\)-congruence between pencils of complex symmetric or skew-symmetric matrices. In this paper the authors prove the following version of this theorem for Hamiltonian or skew-Hamiltonian matrices: Let \(A,B,C,D\) be matrices in the set \(M_{2m}(\mathbb{C})\) of all complex \(2m\times 2m\) matrices such that one of the following conditions is satisfied: (i) all four matrices are Hamiltonian; or (ii) all four matrices are skew-symmetric Hamiltonian; or (iii) \(A\) and \(C\) are Hamiltonian, while \(B\) and \(D\) are skew-symmetric Hamiltonian, or vice versa. Then, if the pencils \(A+ \lambda B\) and \(C+\lambda D\) are strictly equivalent, they are \(J\)-congruent. Moreover, a symplectic counterpart of a result of \textit{H. Xu} [Linear Algebra Appl. 368, 1--24 (2003; Zbl 1025.15016)] concerning the singular value decomposition of a conjugate symplectic matrix is shown: Every symplectic matrix \(S\in M_{2m}(\mathbb{C})\) admits a singular value decomposition \(S=U\Sigma V^*\), where all of the three factors \(U, \Sigma\) and \(V\) are symplectic. Finally the authors discuss some implications which can be derived from \textit{K. Veselić}'s result on definite pairs of Hermitian matrices [Numer. Math. 64, No. 2, 241--269 (1993; Zbl 0805.65038)] in the case of skew-Hermitian matrices: Let \(A\) be a \(J\)-skew-Hermitian matrix such that the Hermitian pair \((iI,iJ)\) is definite. All conjugate symplectic matrices \(Y\) which bring \(A\) through similarity transformations to the special form \(\left(\begin{smallmatrix} D_1 & iD_2\\ -iD_2 & D_1\end{smallmatrix}\right)\) have the same spectral (or Frobenius) condition number.