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

Published:2009-12-04
Printed: Feb 2010
• Chi-Kwong Li,
College of William and Mary, USA
• Yiu-Tung Poon,
Iowa State University, USA
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## Abstract

Let $A$ and $B$ be $n\times n$ complex Hermitian (or real symmetric) matrices with eigenvalues $a_1 \ge \dots \ge a_n$ and $b_1 \ge \dots \ge b_n$. 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 matrices, real symmetric matrices, inertia, rank, multiple eigenvalues
 MSC Classifications: 15A42 - Inequalities involving eigenvalues and eigenvectors 15A57 - Other types of matrices (Hermitian, skew-Hermitian, etc.)