46 1. Preparatory material

Remark 1.3.6. More generally, one can generate an eigenvalue inequal-

ity whenever the intersection numbers of three Schubert varieties of com-

patible dimensions is non-zero; see [HeRo1995]. In fact, this generates

a complete set of inequalities; see [Klyachko]. One can in fact restrict

attention to those varieties whose intersection number is exactly one; see

[KnTaWo2004]. Finally, in those cases, the fact that the intersection is

one can be proven by entirely elementary means (based on the standard

inequalities relating the dimension of two subspaces V, W to their intersec-

tion V ∩ W and sum V + W ); see [BeCoDyLiTi2010]. As a consequence,

the methods in this section can, in principle, be used to derive all possible

eigenvalue inequalities for sums of Hermitian matrices.

Exercise 1.3.4. Verify the inequalities (1.63) and (1.55) by hand in the

case when A and B commute (and are thus simultaneously diagonalisable),

without the use of minimax formulae.

Exercise 1.3.5. Establish the dual Lidskii inequality

λi1 (A + B) + · · · + λik (A + B) ≥ λi1 (A) + · · · + λik (A)

+ λn−k+1(B) + · · · + λn(B)

for any 1 ≤ i1 · · · ik ≤ n and the dual Weyl inequality

λi+j−n(A + B) ≥ λi(A) + λj(B)

whenever 1 ≤ i, j, i + j − n ≤ n.

Exercise 1.3.6. Use the Lidskii inequality to establish the more general

inequality

n

i=1

ciλi(A + B) ≤

n

i=1

ciλi(A) +

n

i=1

ci

∗λi(B)

whenever c1,...,cn ≥ 0, and c1

∗

≥ · · · ≥ cn

∗

≥ 0 is the decreasing rearrange-

ment of c1,...,cn. (Hint: Express ci as the integral of I(ci ≥ λ) as λ

runs from 0 to infinity. For each fixed λ, apply (1.63).) Combine this with

H¨ older’s inequality to conclude the p-Weilandt-Hoffman inequality

(1.65) (λi(A + B) − λi(A))i=1

n

p

n

≤ B

Sp

for any 1 ≤ p ≤ ∞, where

(ai)i=1

n

p

n

:= (

n

i=1

|ai|p)1/p

is the usual

p

norm (with the usual convention that (ai)i=1

n

∞

n

:=

sup1≤i≤p |ai|), and

(1.66) B

Sp

:= (λi(B))i=1

n

p

n

is the p-Schatten norm of B.