304 CHAPTER 11. LIMITS OF VECTORS AND MATRICES
Perhaps this isn’t the first thing you would think of. Of course the ability to get this niceclosed form description of sin(A) was dependent on being able to find the Jordan formalong with a similarity transformation which will yield the Jordan form.
The following corollary is known as the spectral mapping theorem.
Corollary 11.5.3 Let A be an n× n matrix and let ρ (A) < R where for |λ | < R, f (λ ) =∑
∞n=0 anλ
n.Then f (A) is also an n×n matrix and furthermore, σ ( f (A)) = f (σ (A)) . Thusthe eigenvalues of f (A) are exactly the numbers f (λ ) where λ is an eigenvalue of A.Furthermore, the algebraic multiplicity of f (λ ) coincides with the algebraic multiplicityof λ .
All of these things can be generalized to linear transformations defined on infinite di-mensional spaces and when this is done the main tool is the Dunford integral along withthe methods of complex analysis. It is good to see it done for finite dimensional situationsfirst because it gives an idea of what is possible.
11.6 Exercises1. Suppose the migration matrix for three locations is .5 0 .3
.3 .8 0
.2 .2 .7
.
Find a comparison for the populations in the three locations after a long time.
2. Show that if ∑i ai j = 1, then if A = (ai j) , then the sum of the entries of Av equals thesum of the entries of v. Thus it does not matter whether ai j ≥ 0 for this to be so.
3. If A satisfies the conditions of the above problem, can it be concluded that limn→∞ An
exists?
4. Give an example of a non regular Markov matrix which has an eigenvalue equal to−1.
5. Show that when a Markov matrix is non defective, all of the above theory can beproved very easily. In particular, prove the theorem about the existence of limn→∞ An
if the eigenvalues are either 1 or have absolute value less than 1.
6. Find a formula for An where
A =
52 − 1
2 0 −15 0 0 −472 − 1
212 − 5
272 − 1
2 0 −2
Does limn→∞ An exist? Note that all the rows sum to 1. Hint: This matrix is similarto a diagonal matrix. The eigenvalues are 1,−1, 1
2 ,12 .