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

Last updated: 14 October 2014


Let λ1(A) λ2(A) λn(A) denote the (real) eigenvalues of an n-by-n hermitian matrix A. 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 Qd,n denote the set of strictly increasing sequences consisting of d integers chosen from {1,2,,n}. Given a in Qd,n and hermitian matrix A, set Σa(A)= i=1d λai(A). Problem: characterize the set Sdn= { (a,b,c)| 1)a,b,cQd,n, 2)For any three hermitian matricesA,B,C Σa(A)+ Σb(B) Σc(A+B) } . The results above explicitly display certain subsets of Sdn.

In 1962, A. Horn discovered a set Tdn which he conjectured to be a subset of Sdn. The elements (a,b,c) of Tdn are not specified explicitly, but are characterized as sequences whose components satisfy a complex system of linear inequalities. Horn verified that Sdn contains Tdn for all n when d=1,2, or 3, using a technique suggested by Wielandt. It was not indicated whether Tdn contained the previously known results as special cases. In 1962, J. Hersch and B. P. Zwahlen developed a new method for obtaining elements in Sdn. 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 d4, 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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