13.3 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 13.3.1 Define σ
to be the eigenvalues of A. Also,
The number, ρ
is known as the spectral radius of A.
Recall the following symbols and their meaning.
They are respectively the largest and smallest limit points of the sequence
is allowed in the case where the sequence is unbounded. They are also defined as
Thus, the limit of the sequence exists if and only if these are both equal to the same real number.
Also note that the
Lemma 13.3.2 Let J be a p × p Jordan matrix
where each Jk is of the form
in which Nk is a nilpotent matrix having zeros down the main diagonal and ones down the super
where ρ = max
. Here the norm is the operator norm.
Proof: Consider one of the blocks,
is p × p
Now there are p numbers
so you could pick the largest,
so 13.9 is dominated by
The ratio or root test shows that this converges to 0 as n →∞.
What happens when
where C = max
Next let x be an eigenvector for λ,
= 1. Then
The following theorem is due to Gelfand around 1941.
Theorem 13.3.3 (Gelfand) Let A be a complex p×p matrix. Then if ρ is the absolute value of
its largest eigenvalue,
is any norm on ℒ
Proof: First assume
is the operator norm with respect to the usual Euclidean metric on
. Then letting J
denote the Jordan form of A,S−1AS
it follows from Lemma 13.3.2
be the largest eigenvalue of A,
If follows that liminf n→∞
Now by equivalence of norms, if
is any other norm for the set of complex
there exist constants δ,
Δ such that
The limits exist and equal ρ for the ends of the above inequality. Hence, by the squeezing theorem,
ρ = limn→∞
Example 13.3.4 Consider
. Estimate the absolute value of the largest
A laborious computation reveals the eigenvalues are 5, and 10. Therefore, the right answer in
this case is 10. Consider
where the norm is obtained by taking the maximum of all the
absolute values of the entries. Thus
and taking the seventh root of the largest entry gives
Of course the interest lies primarily in matrices for which the exact roots to the characteristic
equation are not known and in the theoretical significance.