The Schubert Calculus and Eigenvalue Inequalities for Sums of Hermitian Matrices

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 14 October 2014

Introduction

Let $${\lambda }_1\left(A\right)\ge {\lambda }_2\left(A\right)\ge \cdots \ge {\lambda }_n\left(A\right)$$ 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 $${\lambda }_1\left(A\right)+{\lambda }_1\left(B\right)\ge {\lambda }_1(A+B)\text{.}$$ A number of generalizations have appeared: $$\begin{array}{cc}{\lambda }_p\left(A\right)+{\lambda }_q\left(B\right)\ge {\lambda }_p+(q-1)(A+B)& \text{Weyl (1912)}\\ {\displaystyle \sum _{i=1}^d{\lambda }_i\left(A\right)+\sum _{i=1}^d{\lambda }_i\left(B\right)\ge \sum _{i=1}^d{\lambda }_i(A+B)}& \text{Fan (1949), Lidskii (1950)}\\ {\displaystyle \sum _{i=1}^d{\lambda }_{p_i}\left(A\right)+\sum _{i=1}^d{\lambda }_i\left(B\right)\ge \sum _{i=1}^d{\lambda }_{p_i}(A+B)}& \text{Wielandt (1955)}\\ {\displaystyle \sum _{i=1}^d{\lambda }_{p_i}\left(A\right)+\sum _{i=1}^d{\lambda }_{q_i}\left(B\right)\ge \sum _{i=1}^d{\lambda }_{p_i+q_i-i}(A+B)}& \text{Thompson (1970)}\end{array}$$ 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 $\{1,2,\dots ,n\}\text{.}$ Given $a$ in $Q_{d,n}$ and hermitian matrix $A,$ set $${\mathrm{\Sigma }}_a\left(A\right)=\sum _{i=1}^d{\lambda }_{a_i}\left(A\right)\text{.}$$ Problem: characterize the set $${}^{•}S_d^n=\left\{(a,b,c)\hspace{0.17em}\right|\hspace{0.17em}\begin{array}{ll}\text{1)}& a,b,c\in Q_{d,n},\\ \text{2)}& \text{For any three hermitian matrices}\hspace{0.17em}A,B,C\\ & {\mathrm{\Sigma }}_a\left(A\right)+{\mathrm{\Sigma }}_b\left(B\right)\ge {\mathrm{\Sigma }}_c(A+B)\end{array}\}\text{.}$$ 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 $(a,b,c)$ 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.

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