The Hua Matrix Equality and Determinantal Inequalities

Grant Winners

Dean

Abstract

The method of iteration is of central importance in matrix computations. A solution to a matrix (or linear dynamical) equation system from real life can be obtained through iterations if the matrix is a contraction (or a contractive mapping). The goal of this project is to study some equalities and inequalities involving contraction matrices. The main tool for the research is the method of Schur complements. The results should lead to a better understanding of the matrix contractions and the usefulness of the method of the Schur complements. One of the celebrated theorems from internationally renowned mathematician Loo-Keng Hua presents a beautiful and important matrix equality involving matrix contractions. We shall revisit Hua's matrix identity by a different approach (that is, the method of Schur complements) and show a larger family of matrix identities as well as determinantal inequalities. In particular, we will extend Hua's results to more general matrices, i.e., the matrices that are not necessarily square. We believe that even more interesting than the proofs we present are the tools we introduce to obtain them | a general result on Schur complements and a generalization of Sylvester's law of inertia on eigenvalues. These are widely useful, and appear to be new in the literature. We use these to provide upper and lower bounds for the determinants of matrix products. The type of results we seek in this project will have far reaching applications in mathematics, statistics, computer science, econometrics, control theory, and engineering.