From symplectic eigenvalues of positive definite matrices to their pseudo-orthogonal eigenvalues

Williamson's theorem states that every real symmetric positive definite matrix A of even order can be brought to diagonal form via a symplectic T-congruence transformation. The diagonal entries of the resulting diagonal form are called the symplectic eigenvalues of A. We point at an analog of this classical result related to Hermitian positive definite matrices, *-congruences, and another class of transformation matrices, namely, pseudo-unitary matrices. This leads to the concept of pseudo-unitary (or pseudo-orthogonal, in the real case) eigenvalues of positive definite matrices.