Page 74 - Matrices theory and applications

3.5. Exercises
57
(a) Let a 1 ≤ ··· ≤ a n be a list of real numbers, with a
n
Define
n
n

l(u):=
a j u j ,
a j
j=1
j=1
n
Let K n be the simplex of IR defined by the constraints u j ≥ 0
L(u):= u j . −1 = a 1 > 0.
for every j =1,... ,n,and u j = 1. Show that there exists
j
an element v ∈ K n that maximizes l + L and minimizes |L − l|
on K n simultaneously.
(b) Deduce that
2
a 1 + a n
max l(u)L(u)= .
u∈K n 2
(c) Let A ∈ HPD n and let a 1 , a n be the smallest and largest
n
eigenvalues of A. Show that for every x ∈ C ,
(a 1 + a n ) 2 4
−1
∗
∗
(x Ax)(x A x) ≤ x .
4a 1 a n
11. (Weyl’s inequalities)
Let A, B be two Hermitian matrices of size n × n whose respective
eigenvalues are α 1 ≤ ··· ≤ α n and β 1 ≤ ··· ≤ β n .Define C = A + B
and let γ 1 ≤ ··· ≤ γ n be its eigenvalues.
(a) Show that α j + β 1 ≤ γ j ≤ α j + β n .
n
(b) Let us recall that if F is a linear subspace of CC ,one writes
∗
R A (F)=max{x Ax | x ∈ F, x 2 =1}.
n
Show that if G, H are two linear subspaces of CC ,then R C (G ∩
H) ≤ R A (G)+ R B (H).
(c) Deduce that if l, m ≥ 1and l +m = k +n (hence l +m ≥ n+1),
then
γ k ≤ α l + β m .
(d) Similarly, show that l + m = k + 1 implies
γ k ≥ α l + β m .
(e) Conclude that the function A → λ k (A) that associates to a Her-
mitian matrix its kth eigenvalue (in increasing order) is Lipschitz
with ratio 1, meaning that
|λ k (B) − λ k (A)|≤ B − A 2 = ρ(B − A)
(see the next chapter for the meaning of the norm M 2 and for
the spectral radius ρ(M)).

Remark: The description of the set of the 3n-tuplets ( α, β, γ)as A
and B run over H n is especially delicate. For a complete historical