Page 91 - Matrices theory and applications

4. Norms
74
(b) Deduce from Corollary 5.5.1 that the induced norm · p is not
strictly convex on M n (IR).
n
5. Let N be a norm on IR .
n
(a) For x ∈ CC , define
!
l "
N 1 (x):=inf |α l |N(x ) ,
l
where the infimum is taken over the set of decompositions x =
l l n
α l x with α l ∈ CC and x ∈ IR . Show that N 1 is a norm on
l
n
n
CC (as a CC-vector space) whose restriction to IR is N. Note:
N 1 is called the complexification of N.
(b) Same question as above for N 2 , defined by
2π

1 iθ
N 2 (x):= [e x]dθ,
2π 0
where

[x]:= N( x) + N( x) .
2
2
(c) Show that N 2 ≤ N 1 .
(d) If N(x)= x 1, show that N 1 (x)= x 1 . Considering then the
vector

1
x = ,
i
show that N 2 = N 1 .
6. (continuation of exercise 5)
n
n
The norms N (on IR )and N 1 (on CC ) lead to induced norms on
M n (IR)and M n(CC), respectively. Show that if M ∈ M n (IR), then
N(M)= N 1 (M). Deduce that Theorem 4.3.1 holds true in M n (IR).
7. Let · be an algebra norm on M n (K)(K = IR or CC), that is, a
norm satisfying AB ≤ A · B . Show that ρ(A) ≤ A for every
A ∈ M n (K).
8. In M n(CC), let D be a diagonalizable matrix and N a nilpotent matrix
that commutes with D.Showthat ρ(D)= ρ(D + N).
9. Let B ∈ M n (CC) be given. Assume that there exists an induced norm
such that B = ρ(B). Let λ be an eigenvalue of maximal modulus
and X a corresponding eigenvector. Show that X does not belong to
the range of B − λI n . Deduce that the Jordan block associated to λ
is diagonal (Jordan reduction is presented in Chapter 6).
10. (continuation of exercise 9)