%0 Journal Article
%T From symplectic eigenvalues of positive definite matrices to their pseudo-orthogonal eigenvalues
%J Computational Mathematics and Computer Modeling with Applications (CMCMA)
%I Shahid Beheshti University
%Z 2783-4859
%A Ikramov, Kh.D.
%A Nazari, Alimohammad
%D 2022
%\ 06/01/2022
%V 1
%N 1
%P 17-20
%! From symplectic eigenvalues of positive definite matrices to their pseudo-orthogonal eigenvalues
%K congruence transformation
%K symplectic matrix
%K pseudo-unitary matrix
%K indices of inertia
%K Schur inequality
%R 10.52547/CMCMA.1.1.17
%X 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.
%U https://cmcma.sbu.ac.ir/article_101991_930629420cb285a6a38484522f0c0cdd.pdf