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.
Ikramov,K and Nazari,A . (2022). From symplectic eigenvalues of positive definite matrices to their pseudo-orthogonal eigenvalues. Computational Mathematics and Computer Modeling with Applications (CMCMA), 1(1), 17-20. doi: 10.52547/CMCMA.1.1.17
MLA
Ikramov,K , and Nazari,A . "From symplectic eigenvalues of positive definite matrices to their pseudo-orthogonal eigenvalues", Computational Mathematics and Computer Modeling with Applications (CMCMA), 1, 1, 2022, 17-20. doi: 10.52547/CMCMA.1.1.17
HARVARD
Ikramov K, Nazari A. (2022). 'From symplectic eigenvalues of positive definite matrices to their pseudo-orthogonal eigenvalues', Computational Mathematics and Computer Modeling with Applications (CMCMA), 1(1), pp. 17-20. doi: 10.52547/CMCMA.1.1.17
CHICAGO
K Ikramov and A Nazari, "From symplectic eigenvalues of positive definite matrices to their pseudo-orthogonal eigenvalues," Computational Mathematics and Computer Modeling with Applications (CMCMA), 1 1 (2022): 17-20, doi: 10.52547/CMCMA.1.1.17
VANCOUVER
Ikramov K, Nazari A. From symplectic eigenvalues of positive definite matrices to their pseudo-orthogonal eigenvalues. Computational Mathematics and Computer Modeling with Applications (CMCMA). 2022;1(1):17-20. doi: 10.52547/CMCMA.1.1.17
