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
).
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