The Jordan cononical form of a product of a Hermitian and a positive semidefinite matrix

Abstract

A given square complex matrix C is the product of a positive semidefinite matrix A and a Hermitian matrix B if and only if C 2 is diagonalizable and has nonnegative eigenvalues. This condition is equivalent to requiring that C have real eigenvalues and a Jordan canonical form that is diagonal except for r copies of a 2-by-2 nilpotent Jordan block. We show that r is bounded from above by the rank of A, the nullity of A, and both the positive and negative inertia of B. It follows that a product of two positive semidefinite matrices is diagonalizable and has nonnegative eigenvalues, a result that leads to a characterization of the possible concanonical forms of a positive semidefinite matrix.