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

Abstract

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. Copyright c (cid:13) 2022 Shahid Beheshti University.