Page 92 - Matrices theory and applications

Conversely, show that if the Jordan blocks of B associated to the
eigenvalues of maximal modulus of B are diagonal, then there exists
n
anormon CC such that, using the induced norm, ρ(B)= B .
11. Here is another proof of Theorem 4.2.1. Let K = IR or CC, A ∈
M n (K), and let N be a norm on K .If > 0, we define for all
n
x ∈ K
n −k k 4.6. Exercises 75
x := (ρ(A)+ ) N(A x).
k∈IN
(a) Show that this series is convergent (use Corollary 4.4.1).
n
(b) Show that · is a norm on K .
(c) Show that for the induced norm, A ≤ ρ(A)+ .
12. A matrix norm · on M n (CC)is said to be unitarily invariant if
UAV = A for every A ∈ M n (CC) and all unitary matrices U, V .
(a) Find, among the most classical norms, two examples of unitarily
invariant norms.
(b) Given a unitarily invariant norm, show that there exists a norm
n
N on IR such that
A = N(s 1 (A),... ,s n (A)),
where the s j (A)’s, the eigenvalues of H in the polar decompo-
sition A = QH (see Chapter 7 for this notion), are called the
singular values of A.
13. (R. Bhatia [5]) Suppose we are given a norm · on M n(CC)that
is unitarily invariant (see the previous exercise). If A ∈ M n (CC), we
denote by D(A) the diagonal matrix obtained by keeping only the
a jj and setting all the other entries to zero. If σ is a permutation,
σ
we denote by A the matrix whose entry of index (j, k)equals a jk if
k = σ(j), and zero otherwise. For example, A id = D(A), where id is
the identity permutation. If r is an integer between 1 − n and n − 1,
we denote by D r (A) the matrix whose entry of index (j, k)equals a jk
if k − j = r, and zero otherwise. For example, D 0 (A)= D(A).
(a) Let ω =exp(2iπ/n)and let U be the diagonal matrix whose
diagonal entries are the roots of unity 1,ω,... ,ω n−1 . Show that
n−1
1 ∗j j
D(A)= U AU .
n
j=0
Deduce that D(A) ≤ A .
σ
(b) Show that A ≤ A for every σ ∈S n .Observe that P =
I n for every permutation matrix P. Show that M ≤ I n
for every bistochastic matrix M (see Section 5.5 for this notion).