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Sum of Hermitian Matrices with Given Eigenvalues: Inertia, Rank, and Multiple Eigenvalues

Part of: Basic linear algebra

Published online by Cambridge University Press:  20 November 2018

Chi-Kwong Li  and
Yiu-Tung Poon
Chi-Kwong Li
Affiliation:
Department of Mathematics, College of William and Mary, Williamsburg, VA 23185, USA and University of Hong Kong, e-mail: ckli@math.wm.edu
Yiu-Tung Poon
Affiliation:
Department of Mathematics, Iowa State University, Ames, IA 50011, USA, e-mail: ytpoon@iastate.edu
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Abstract

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Let $A$ and $B$ be $n\,\times \,n$ complex Hermitian (or real symmetric) matrices with eigenvalues ${{a}_{1}}\,\ge \,\cdots \,\ge \,{{a}_{n}}$ and ${{b}_{1}}\,\ge \,\cdots \,\ge \,{{b}_{n}}$. All possible inertia values, ranks, and multiple eigenvalues of $A\,+\,B$ are determined. Extension of the results to the sum of $k$ matrices with $k\,>\,2$ and connections of the results to other subjects such as algebraic combinatorics are also discussed.

Keywords

complex Hermitian matricesreal symmetric matricesinertiarankmultiple eigenvalues

MSC classification

Secondary: 15A42: Inequalities involving eigenvalues and eigenvectors
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 62 , Issue 1 , 01 February 2010 , pp. 109 - 132
Copyright
Copyright © Canadian Mathematical Society 2010

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