Last updated: 14 October 2014

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 $${}^{\u2022}S_{d}^{n}=\left\{(a,b,c)\hspace{0.17em}\right|\hspace{0.17em}\begin{array}{ll}\text{1)}& a,b,c\in {Q}_{d,n},\\ \text{2)}& \text{For any three hermitian matrices}\hspace{0.17em}A,B,C\\ & {\mathrm{\Sigma}}_{a}\left(A\right)+{\mathrm{\Sigma}}_{b}\left(B\right)\ge {\mathrm{\Sigma}}_{c}(A+B)\end{array}\}\text{.}$$ The results above explicitly display certain subsets of ${}^{\u2022}S_{d}^{n}\text{.}$

In 1962, A. Horn discovered a set ${}^{\u2022}T_{d}^{n}$ which he conjectured to be a subset of ${}^{\u2022}S_{d}^{n}\text{.}$ The elements $(a,b,c)$ of ${}^{\u2022}T_{d}^{n}$ are not specified explicitly, but are characterized as sequences whose components satisfy a complex system of linear inequalities. Horn verified that ${}^{\u2022}S_{d}^{n}$ contains ${}^{\u2022}T_{d}^{n}$ for all $n$ when $d=1,2,$ or $3,$ using a technique suggested by Wielandt. It was not indicated whether ${}^{\u2022}T_{d}^{n}$ contained the previously known results as special cases. In 1962, J. Hersch and B. P. Zwahlen developed a new method for obtaining elements in ${}^{\u2022}S_{d}^{n}\text{.}$ 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.

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