Eigenvalues for Sums of Hermitian Matrices

Abstract

In this thesis we explore how the eigenvalues of nxn Hermitian matrices A,B relate to the eigenvalues of their sum C=A+B. We mainly focus on inequalities bounding sums of r eigenvalues for C by sums of r eigenvalues for A with r eigenvalues for B, for some r less than n. We begin by using linear algebra to establish some classical results, including a result by R.C. Thompson that allows us to reformulate the eigenvalue problem in terms of nonempty intersections in the Grassmannian manifold of r-planes in complex n-dimensional space. In particular, every nonempty triple intersection of Schubert varieties in a Grassmannian yields an eigenvalue inequality. Such nonempty intersections correspond to nonzero cup products in the cohomology ring of the Grassmannian, and consequently, to nonzero Littlewood-Richardson coefficients. The Littlewood-Richardson rules provide us with an efficient method of detecting when these coefficients are nonzero, and hence of finding eigenvalue inequalities which necessarily hold for all nxn Hermitian matrices A,B,C=A+B. For the remainder of this thesis, we turn our attention to particular inequalities of the above form that Alfred Horn conjectured would completely determine the possible eigenvalues of A,B,C=A+B. Horn's conjecture, formulated in 1962, was resolved in the affirmative during the late 1990's in celebrated work of A. Knutson and T. Tao, building on results of A. Klyachko and others. We will develop the connection between Horn's inequalities and the earlier parts of this thesis. In particular, we will see that each Horn inequality corresponds to a nonzero cup product that lies in the top degree cohomology group of the Grassmannian. An alternate formulation of Horn's Theorem shows that indices yield a Horn inequality if and only if certain associated partitions occur as the eigenvalues for some rxr Hermitian matrices A, B, C=A+B. We will prove that when r=n-2 there are necessarily diagonal rxr matrices satisfying this condition.