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 13.3.1Define 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 snu→p∞ an, lim ni→nf∞an

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 an ≡ lim (sup {ak : k ≥ n}),
n→∞ n→ ∞
lim nin→f∞ an ≡ nl→im∞ (inf{ak : k ≥ n}).

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 13.3.2Let J be a p × p Jordan matrix

( )
J1
J = || .. ||
( . )
Js

where each J_{k}is of the form

Jk = λkI + Nk

in which N_{k}is a nilpotent matrix having zeros down the main diagonal and ones down the superdiagonal. Then

n 1∕n
nli→m∞ ||J || = ρ

where ρ = max

{|λk|,k = 1,...,n}

. Here the norm is the operator norm.

Proof: Consider one of the blocks,

|λk|

< ρ. Here J_{k} is p × p.

∑p ( )
1-Jnk = -1- n N ikλnk−i
ρn ρn i=0 i

Then

∥ ∥ p ( ) | |
∥∥-1- n∥∥ ∑ n ∥∥ i∥∥ |λnk−i|-1-
∥ρnJ k∥ ≤ i Nk ρn−i ρi (13.9)
i=0

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 ofits largest eigenvalue,

lim ||An ||1∕n = ρ.
n→∞

Here

||⋅||

is any norm on ℒ

n n
(ℂ ,ℂ )

.

Proof:First assume

||⋅||

is the operator norm with respect to the usual Euclidean metric on
ℂ^{n}. Then letting J denote the Jordan form of A,S^{−1}AS = J, it follows from Lemma 13.3.2

and taking the seventh root of the largest entry gives

ρ (A ) ≈ 80156251∕7 = 9.68895123671.

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.