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

http://dx.doi.org/10.4153/CJM-2010-007-2
Canad. J. Math. 62(2010), 109-132

  Published:2009-12-04
 Printed: Feb 2010

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.