Last updated: 14 October 2014
Let denote the (real) eigenvalues of an hermitian matrix Given three such matrices it is a simple fact that A number of generalizations have appeared: These results may be viewed in the context of the following general problem. Let denote the set of strictly increasing sequences consisting of integers chosen from Given in and hermitian matrix set Problem: characterize the set The results above explicitly display certain subsets of
In 1962, A. Horn discovered a set which he conjectured to be a subset of The elements of are not specified explicitly, but are characterized as sequences whose components satisfy a complex system of linear inequalities. Horn verified that contains for all when or using a technique suggested by Wielandt. It was not indicated whether contained the previously known results as special cases. In 1962, J. Hersch and B. P. Zwahlen developed a new method for obtaining elements in 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 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.