Distributions of eigenvalues and inertias of some block Hermitian matrices consisting of orthogonal projectors

Abstract

A complex square matrix A is called an orthogonal projector if A ² = A = A*, where A* is the conjugate transpose of A. In this article, we first give some formulas for calculating the distributions of real eigenvalues of a linear combination of two orthogonal projectors. Then, we establish various expansion formulas for calculating the inertias, ranks and signatures of some 2 × 2 and 3 × 3, as well as k × k block Hermitian matrices consisting of two orthogonal projectors. Many applications of the formulas are presented in characterizing interval distributions of numbers of eigenvalues, and nonsingularity of these block Hermitian matrices. In addition, necessary and sufficient conditions are given for various equalities and inequalities of these block Hermitian matrices to hold.

Keywords:

• orthogonal projector
• Hermitian matrix
• block matrix
• inertia
• rank
• signature
• matrix equality
• matrix inequality
• distribution of eigenvalues
• Löwner partial ordering

AMS Subject Classifications::

• 15A03
• 15A09
• 15A27
• 15B57

Acknowledgements

We are grateful to the two anonymous referees and the handling editor for their careful reading and helpful comments and suggestions, which helped to greatly improve the presentation of this article.