Shahid Beheshti UniversityComputational Mathematics and Computer Modeling with Applications (CMCMA)2783-48591120220601From symplectic eigenvalues of positive definite matrices to their pseudo-orthogonal eigenvalues172010199110.52547/CMCMA.1.1.17ENKh.D.IkramovaMoscow Lomonosov State University, Moscow, RussiaAlimohammadNazariJournal Article20211223Williamson'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.https://cmcma.sbu.ac.ir/article_101991_930629420cb285a6a38484522f0c0cdd.pdf