Diagonalization of Hermitian matrices

Joint Approximate Diagonalization of Positive Deenite Hermitian Matrices

This paper provides an iterative algorithm to jointly approximately di-agonalize K Hermitian positive deenite matrices ? 1 , : : : , ? K. Speciically it calculates the matrix B which minimizes the criterion P K k=1 n k log det diag(BC k B) ? log det(BC k B)], n k being positive numbers, which is a measure of the deviation from diagonality of the matrices BC k B. The convergence of the algorithm...

Joint Approximate Diagonalization of Positive Definite Hermitian Matrices

Congruence of Hermitian Matrices by Hermitian Matrices

Two Hermitian matrices A, B ∈ Mn(C) are said to be Hermitian-congruent if there exists a nonsingular Hermitian matrix C ∈ Mn(C) such that B = CAC. In this paper, we give necessary and sufficient conditions for two nonsingular simultaneously unitarily diagonalizable Hermitian matrices A and B to be Hermitian-congruent. Moreover, when A and B are Hermitian-congruent, we describe the possible iner...

A block variational procedure for the iterative diagonalization of non-Hermitian random-phase approximation matrices.

We present a technique for the iterative diagonalization of random-phase approximation (RPA) matrices, which are encountered in the framework of time-dependent density-functional theory (TDDFT) and the Bethe-Salpeter equation. The non-Hermitian character of these matrices does not permit a straightforward application of standard iterative techniques used, i.e., for the diagonalization of ground...

Diagonality Measures of Hermitian Positive-Definite Matrices with Application to the Approximate Joint Diagonalization Problem

In this paper, we introduce properly-invariant diagonality measures of Hermitian positive-definite matrices. These diagonality measures are defined as distances or divergences between a given positive-definite matrix and its diagonal part. We then give closed-form expressions of these diagonality measures and discuss their invariance properties. The diagonality measure based on the log-determin...