Linear Algebra, Theory And Applications, 2012a

348 NORMS FOR FINITE DIMENSIONAL VECTOR SPACES
14.3 The Spectral Radius
Even though it is in general impractical to compute the Jordan form, its existence is all that
is needed in order to prove an important theorem about something which is relatively easy
to compute. This is the spectral radius of a matrix.
Definition 14.3.1 Define σ (A) to be the eigenvalues of A. Also,
ρ (A) ≡ max (|λ| : λ ∈ σ (A))
The number, ρ (A) is known as the spectral radius of A.
Recall the following symbols and their meaning.
lim sup a n , lim inf a n
n→∞
n→∞
They are respectively the largest and smallest limit points of the sequence {a n } where ±∞
is allowed in the case where the sequence is unbounded. They are also defined as
lim sup a n
n→∞
≡ lim k : k ≥ n}) ,
n→∞
lim inf n
n→∞
≡ lim k : k ≥ n}) .
n→∞
Thus, the limit of the sequence exists if and only if these are both equal to the same real
number.
. ..
J s
Lemma 14.3.2 Let J be a p × p Jordan matrix
⎛
⎜
J = ⎝
J 1
⎞
⎟
⎠
where each J k is of the form
J k = λ k I + N k
in which N k is a nilpotent matrix having zeros down the main diagonal and ones down the
super diagonal. Then
lim ||J n || 1/n = ρ
n→∞
where ρ =max{|λ k | ,k =1,...,n}. Here the norm is defined to equal
||B|| =max{|B ij | ,i,j} .
Proof: Suppose first that ρ ≠0. First note that for this norm, if B,C are p×p matrices,
||BC|| ≤ p ||B|| ||C||
which follows from a simple computation. Now
⎛
(λ 1 I + N 1 ) n ⎞
||J n || 1/n ⎜
=
⎝
. ..
⎟
⎠
∣∣
(λ s I + N s ) n ∣∣
1/n