The inertia of a Hermitian matrix having prescribed complementary principal submatrices

Rank and inertia of submatrices of the Moore-Penrose inverse of a Hermitian matrix

Off-diagonal Submatrices of a Hermitian Matrix

A necessary and sufficient condition is given to a p × q complex matrix X to be an off-diagonal block of an n × n Hermitian matrix C with prescribed eigenvalues (in terms of the eigenvalues of C and singular values of X). The proof depends on some recent breakthroughs in the study of spectral inequalities on the sum of Hermitian matrices by Klyachko and Fulton. Some interesting geometrical prop...

Ela Rank and Inertia of Submatrices of the Moore–penrose Inverse of a Hermitian Matrix

Closed-form formulas are derived for the rank and inertia of submatrices of the Moore–Penrose inverse of a Hermitian matrix. A variety of consequences on the nonsingularity, nullity and definiteness of the submatrices are also presented.

Submodular spectral functions of principal submatrices of a hermitian matrix, extensions and applications

We extend the multiplicative submodularity of the principal determinants of a nonnegative definite hermitian matrix to other spectral functions. We show that if f is the primitive of a function that is operator monotone on an interval containing the spectrum of a hermitian matrix A, then the function I 7→ trf(A[I]) is supermodular, meaning that trf(A[I])+trf(A[J ]) 6 trf(A[I∪ J ]) + trf(A[I ∩ J...

The reconstruction of an hermitian toeplitz matrix with prescribed eigenpairs

Toeplitz matrices have been found important applications in bioinformatics and computational biology [5-9, 11-12]. In this paper we concern the reconstruction of an hermitian Toeplitz matrices with prescribed eigenpairs. Based on the fact that every centrohermitian matrix can be reduced to a real matrix by a simple similarity transformation, we first consider the eigenstructure of hermitian Toe...