Positivity of a Hadamard Product

TL;DR

This paper establishes an eigenvalue lower bound for the Hadamard product of positive semidefinite matrices, revealing unique spectral properties and applications in signal processing.

Contribution

It introduces a new eigenvalue lower bound for Hadamard products that depends on matrix rank, condition number, and principal submatrix eigenvalues.

Findings

Eigenvalue lower bound depends on rank, condition number, and principal submatrix eigenvalues.

Numerical examples illustrate the bound's effectiveness.

Applications discussed in array signal processing and matrix time series analysis.

Abstract

A notable difference between the ordinary and Hadamard products is that the Hadamard product of two singular positive semidefinite matrices can be nonsingular, and one of the factors can even be indefinite. We present an eigenvalue lower bound for a Hadamard product that depends on the rank, effective condition number, and diagonal entries of one factor, and the smallest eigenvalues of certain principal submatrices of the other factor. We give numerical examples and discuss its applications in array signal processing and matrix time series analysis.