Document Type : Regular paper

Authors

• Kh.D. Ikramov
• Alimohammad Nazari

aMoscow Lomonosov State University, Moscow, Russia

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.

Keywords

• congruence transformation
• symplectic matrix
• pseudo-unitary matrix
• indices of inertia
• Schur inequality