Page 87 - Matrices theory and applications

4. Norms
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4.4 A Lemma about Banach Algebras
Definition 4.4.1 A normed algebra is a K-algebra endowed with a norm
satisfying xy ≤ x y . Such a norm is called an algebra norm.When
a normed algebra is complete (which is always true in finite dimension), it
is called a Banach algebra.
Lemma 4.4.1 Let A be a normed algebra and let x ∈A. The sequence
m 1/m
u m := x converges to its infimum, denoted by r(x). Additionally,
if K = CC,andif A has a unit element and is complete, then 1/r(x) is
the radius of the largest open ball B(0; R) such that e − zx is invertible for
every z ∈ B(0; R).
Of course, one may apply the lemma to A = M n (CC) endowed with
a matrix norm. One then has r(x)= ρ(x), because e − zx = I − zA is
invertible, provided that z is not the inverse of an eigenvalue. In the case
K = IR, one uses an auxiliary norm N that is the restriction to M n (IR)of
an induced norm on M n (CC). Since · and N are equivalent, one simply
writes
m 1/m
m 1/m
m 1/m
ρ(A)= ρ(A ) ≤ A ≤ C 1/m N(A ) .
The latter sequence converges to ρ(A) from the lemma, which implies the
convergence of the former. We thus have the following result.
Proposition 4.4.1 If A ∈ M n(K),then
m 1/m
ρ(A) = lim A
m→∞
for every matrix norm.
Proof

Convergence. The result is trivial if x m = 0 for some exponent. In the
opposite case, we use the following inequalities, which come directly
from the definition:
p a
r
x ap+r ≤ x x , ∀a, p, r ∈ IN.
We then define
1
m
v m = log x =log u m .
m
Let us fix an integer p and perform Euclidean division of m by p:
m = ap + r with 0 ≤ r ≤ p − 1. This yields
apv p + rv r
v ap+r ≤ .
ap + r
As m, hence a, tends to infinity, the right-hand side converges,
because rv r remains bounded:

lim sup v m ≤ v p .