## The Schubert Calculus and Eigenvalue Inequalities for Sums of Hermitian Matrices

Last updated: 14 October 2014

## Introduction

Let $λ1(A)≥ λ2(A)≥⋯≥ λn(A)$ denote the (real) eigenvalues of an $n\text{-by-}n$ hermitian matrix $A\text{.}$ Given three such matrices $A,B,A+B,$ it is a simple fact that $λ1(A)+ λ1(B)≥ λ1(A+B).$ A number of generalizations have appeared: $λp(A)+ λq(B)≥ λp+(q-1)(A+B) Weyl (1912) ∑i=1dλi(A)+ ∑i=1dλi(B)≥ ∑i=1dλi(A+B) Fan (1949), Lidskii (1950) ∑i=1dλpi(A)+ ∑i=1dλi(B)≥ ∑i=1dλpi(A+B) Wielandt (1955) ∑i=1dλpi(A)+ ∑i=1dλqi(B)≥ ∑i=1dλpi+qi-i(A+B) Thompson (1970)$ These results may be viewed in the context of the following general problem. Let ${Q}_{d,n}$ denote the set of strictly increasing sequences consisting of $d$ integers chosen from $\left\{1,2,\dots ,n\right\}\text{.}$ Given $a$ in ${Q}_{d,n}$ and hermitian matrix $A,$ set $Σa(A)= ∑i=1d λai(A).$ Problem: characterize the set $Sdn•= { (a,b,c) | 1)a,b,c∈Qd,n, 2)For any three hermitian matrices A,B,C Σa(A)+ Σb(B)≥ Σc(A+B) } .$ The results above explicitly display certain subsets of ${}^{•}S_{d}^{n}\text{.}$

In 1962, A. Horn discovered a set ${}^{•}T_{d}^{n}$ which he conjectured to be a subset of ${}^{•}S_{d}^{n}\text{.}$ The elements $\left(a,b,c\right)$ of ${}^{•}T_{d}^{n}$ are not specified explicitly, but are characterized as sequences whose components satisfy a complex system of linear inequalities. Horn verified that ${}^{•}S_{d}^{n}$ contains ${}^{•}T_{d}^{n}$ for all $n$ when $d=1,2,$ or $3,$ using a technique suggested by Wielandt. It was not indicated whether ${}^{•}T_{d}^{n}$ contained the previously known results as special cases. In 1962, J. Hersch and B. P. Zwahlen developed a new method for obtaining elements in ${}^{•}S_{d}^{n}\text{.}$ Using these methods, Zwahlen (1964) confirmed Horn's results. R. C. Thompson (1970) showed that the key idea underlying Horn's methods, the "pushing" lemma, could be adapted to the Hersch-Zwahlen technique, greatly simplifying their proofs.

The present work introduces two new tools for use in conjunction with the Hersch-Zwahlen-Thompson methods:

 1) The Schubert Calculus of classical algebraic geometry, 2) The Littlewood-Richardson rule for the multiplication of Schur functions in the algebra of symmetric polynomials.

A reformulation of the Littlewood-Richardson rule is used to prove a generalization of the "pushing" lemma; this in turn leads to a proof of Horn's conjecture for $d\le 4,$ and makes a complete proof contingent only on an easily stated, plausible lemma.

## Notes and References

This is an excerpt from Steven Andrew Johnson's 1979 dissertation The Schubert Calculus and Eigenvalue Inequalities for Sums of Hermitian Matrices.