Page 84 - Matrices theory and applications

4.3. An Interpolation Inequality
Theorem 4.2.1 For every B ∈ M n (CC) and all > 0, there exists a norm
n
on CC such that for the induced norm
In other words, ρ(B) is the infimum of B ,as · ranges over the set
of matrix norms.
Proof B ≤ ρ(B)+ . 67
From Theorem 2.7.1 there exists P ∈ GL n (CC) such that T := PBP −1
is upper triangular. From Proposition 4.1.7, one has
inf B =inf PBP −1 =inf T ,
where the infimum is taken over the set of induced norms. Since B and
T have the same spectra, hence the same spectral radius, it is enough to
prove the theorem for upper triangular matrices.
For such a matrix T , Proposition 4.1.7 still gives
inf T ≤ inf{ QTQ −1 2 ; Q ∈ GL n (CC)}.
2
Let us now take Q(µ) = diag(1,µ,µ ,... ,µ n−1 ). The matrix Q(µ)TQ(µ) −1
is upper triangular, with the same diagonal as that of T . Indeed, the entry
with indices (i, j) becomes µ i−j t ij . Hence,

lim Q(µ)TQ(µ) −1
µ→∞
is simply the matrix D = diag(t 11 ,... ,t nn ). Since · 2 is continuous (as
is every norm), one deduces
inf T ≤ lim Q(µ)TQ(µ) −1 2 = D 2 = ρ(D D)= max |t jj | = ρ(T ).
∗
µ→∞
Remark: The theorem tells us that ρ(A)=Λ(A), where

Λ(A):= inf A ,
the infimum being taken over the set of matrix norms. The first part of the
proof tells us that ρ and Λ coincide on the set of diagonalizable matrices,
which is a dense subset of M n (CC). But this is insufficient to conclude,
since Λ is a priori only upper semicontinuous, as the infimum of continuous
functions. The continuity of Λ is actually a consequence of the theorem.

4.3 An Interpolation Inequality

Theorem 4.3.1 (case K = CC) Let · p be the norm on M n (CC) induced
n
p
by the norm l on CC . The function
1/p → log A p ,
[0, 1] → IR,