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4
Norms
4.1 A Brief Review
n
In this Chapter, the field K will always be IR or CC and E will denote K .
If A ∈ M n(K), the spectral radius of A, denoted by ρ(A), is defined as
the largest modulus of the eigenvalues of A:
ρ(A)= max{|λ|; λ ∈ Sp(A)}.
When K = IR, one takes into account the complex eigenvalues when
computing ρ(A).
The scalar (if K = IR) or Hermitian (if K = CC) product on E is denoted
by (x, y):= x j ¯ j . The vector space E is endowed with various norms,
y
j
pairwise equivalent since E has finite dimension (Proposition 4.1.3 below).
p
Among these, the most used norms are the l norms:
1/p
p
x p = |x j | , x ∞ =max |x j |.
j
j
Proposition 4.1.1 For 1 ≤ p ≤∞,the map x → x p is a norm on E.
In particular, one has Minkowski’s inequality
x + y p ≤ x p + y p. (4.1)
