44 1. Preparatory material

Specialising Proposition 1.3.4 to the case when V is a coordinate sub-

space (i.e., the span of k of the basis vectors e1,...,en), we conclude the

Schur-Horn inequalities

λn−k+1(A) + · · · + λn(A) ≤ ai1i1 + · · · + aikik

≤ λ1(A) + · · · + λk(A)

(1.60)

for any 1 ≤ i1 · · · ik ≤ n, where a11,a22,...,ann are the diagonal

entries of A.

Exercise 1.3.2. Show that the inequalities (1.60) are equivalent to the

assertion that the diagonal entries diag(A) = (a11,a22,...,ann) lies in the

permutahedron of λ1(A),...,λn(A), defined as the convex hull of the n!

permutations of (λ1(A),...,λn(A)) in

Rn.

Remark 1.3.5. It is a theorem of Schur and Horn [Ho1954] that these are

the complete set of inequalities connecting the diagonal entries diag(A) =

(a11,a22,...,ann) of a Hermitian matrix to its spectrum. To put it an-

other way, the image of any coadjoint orbit OA := {UAU

∗

: U ∈ U(n)}

of a matrix A with a given spectrum λ1,...,λn under the diagonal map

diag : A → diag(A) is the permutahedron of λ1,...,λn. Note that the

vertices of this permutahedron can be attained by considering the diagonal

matrices inside this coadjoint orbit, whose entries are then a permutation of

the eigenvalues. One can interpret this diagonal map diag as the moment

map associated with the conjugation action of the standard maximal torus

of U(n) (i.e., the diagonal unitary matrices) on the coadjoint orbit. When

viewed in this fashion, the Schur-Horn theorem can be viewed as the special

case of the more general Atiyah convexity theorem [At1982] (also proven

independently by Guillemin and Sternberg [GuSt1982]) in symplectic ge-

ometry. Indeed, the topic of eigenvalues of Hermitian matrices turns out to

be quite profitably viewed as a question in symplectic geometry (and also

in algebraic geometry, particularly when viewed through the machinery of

geometric invariant theory).

There is a simultaneous generalisation of Theorem 1.3.2 and Proposition

1.3.4:

Exercise 1.3.3 (Wielandt minimax formula). Let 1 ≤ i1 · · · ik ≤ n

be integers. Define a partial flag to be a nested collection V1 ⊂ · · · ⊂ Vk

of subspaces of

Cn

such that dim(Vj) = ij for all 1 ≤ j ≤ k. Define

the associated Schubert variety X(V1,...,Vk) to be the collection of all k-

dimensional subspaces W such that dim(W ∩ Vj) ≥ j. Show that for any

n × n matrix A,

λi1 (A) + · · · + λik (A) = sup

V1,...,Vk

inf

W ∈X(V1,...,Vk)

tr(A

W

).