Page 83 - Matrices theory and applications

4. Norms
66
k
B

B
plete, the series
converges. Furthermore, (I n − B)
k≤N
k
N+1
I n − B
, which tends to I n . The sum of the series is thus the inverse of
I n − B. One has, moreover,
1

k
−1
(I n − B)
B =
.
≤
1 − B
k
One can also deduce Proposition 4.1.5 from the following statement. k =
Proposition 4.1.6 For every induced norm, one has
ρ(A) ≤ A .
Proof
The case K = CC is easy, because there exists an eigenvector X ∈ E
associated to an eigenvalue of modulus ρ(A):
ρ(A) X = λX = AX ≤ A X .
If K = IR, one needs a more involved trick.
n
Let us choose a norm on CC and let us denote by N the induced norm
on M n(CC). We still denote by N its restriction to M n (IR); it is a norm.
Since this space has finite dimension, any two norms are equivalent: There
exists C> 0 such that N(B) ≤ C B for every B in M n (IR). Using the
result already proved in the complex case, one has for every m ∈ IN that
m
m
m
m
ρ(A) m = ρ(A ) ≤ N(A ) ≤ C A ≤ C A .
Taking the mth root and letting m tend to infinity, and noticing that C 1/m
tends to 1, one obtains the announced inequality.
In general, the equality does not hold. For example, if A is nilpotent
though nonzero, one has ρ(A)= 0 < A for every matrix norm.
n
Proposition 4.1.7 Let · be a norm on K and P ∈ GL n (K).Hence,
n
N(x):= Px defines a norm on K . Denoting still by · and N the
n
induced norms on K , one has N(A)= PAP −1 .
Proof
Using the change of dummy variable y = Px,we have
PAx PAP −1 y −1
N(A)=sup =sup = PAP .
x =0 Px y =0 y
4.2 Householder’s Theorem
Householder’s theorem is a kind of converse of the inequality ρ(B) ≤ B .