Page 95 - Matrices theory and applications

4. Norms
78
We assume in this exercise that δ> 0and ≤ δ/4n.
(a) Show that each Gershgorin disk D j contains exactly one
eigenvalue of A.
ρ
(b) Let ρ> 0 be a real number. Show that A , obtained by multi-
plying the ith row of A by ρ and the ith column by 1/ρ,has the
same eigenvalues as A.
ρ
(c) Choose ρ =2 /δ. Show that the ith Gershgorin disk of A con-
tains exactly one eigenvalue. Deduce that the eigenvalues of A
aresimpleand that
2n 2
d(Sp(A), diag(A)) ≤ ,
δ
where diag(A)= {a 11 ,... ,a nn }.
19. Let A ∈ M n (CC) be a diagonalizable matrix:
A = S diag(d 1 ,... ,d n )S −1 .
Let · be an induced norm for which D =max j |d j | holds, where
D := diag(d 1 ,... ,d n ). Show that for every E ∈ M n (CC)and forevery
eigenvalue λ of A + E, there exists an index j such that
|λ − d j |≤ S · S −1 · E .
20. Let A ∈ M n (K), with K = IR or CC. Give another proof, using
the Cauchy–Schwarz inequality, of the following particular case of
Theorem 4.3.1:
1/2 1/2
A 2 ≤ A A .
1 ∞
21. Show that if A ∈ M n (CC)is normal, then ρ(A)= A 2. Deduce that
if A and B are normal, ρ(AB) ≤ ρ(A)ρ(B).
n
22. Let N 1 and N 2 be two norms on CC .Denote by N 1 and N 2 the
induced norms on M n (CC). Let us define
N 1 (x) N 2 (x)
R := max , S := max .
x =0 N 2 (x) x =0 N 1 (x)
(a) Show that
N 1 (A) N 2 (A)
max = RS =max .
A =0 N 2 (A) A =0 N 1 (A)
(b) Deduce that if N 1 = N 2 ,then N 2 /N 1 is constant.
(c) Show that if N 1 ≤N 2 ,then N 2 /N 1 is constant and therefore
N 2 = N 1 .

23. (continuation of exercise 22)
Let · be an algebra norm on M n (CC). If y ∈ CC n is nonzero, we
∗
define x y := xy .