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

Document Type : Regular paper


aMoscow Lomonosov State University, Moscow, Russia


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.


Volume 1, Issue 1 - Serial Number 1
January 2022
Pages 17-20
  • Receive Date: 23 December 2021
  • Revise Date: 27 December 2021
  • Accept Date: 31 December 2021
  • First Publish Date: 01 January 2022