Theory of Statistics - George Mason University

812 0 Statistical Mathematics
The relationship (0.3.68) allows us to prove properties 1 and 4.
The one property of square positive matrices that does not carry over to
square irreducible nonnegative matrices is that r = ρ(A) is the only eigenvalue
on the spectral circle of A. For example, the small irreducible nonnegative
matrix
0 1
A =
1 0
has eigenvalues 1 and −1, and so both are on the spectral circle.
It turns out, however, that square irreducible nonnegative matrices that
have only one eigenvalue on the spectral circle also have other interesting
properties that are important, for example, in Markov chains. We therefore
give a name to the property:
A square irreducible nonnegative matrix is said to be primitive if it
has only one eigenvalue on the spectral circle.
In modeling with Markov chains and other applications, the limiting behavior
of A k is an important property.
If A is a primitive matrix, then we have the useful result
k A
lim = vw
k→∞ ρ(A)
T , (0.3.70)
where v is an eigenvector of A associated with ρ(A) and w is an eigenvector
of A T associated with ρ(A), and w and v are scaled so that w T v = 1. (Such
eigenvectors exist because ρ(A) is a simple eigenvalue. They also exist because
they are both positive. Note that A is not necessarily symmetric, and so its
eigenvectors may include imaginary components; however, the eigenvectors
associated with ρ(A) are real, and so we can write w T instead of w H .)
To see equation (0.3.70), we consider A − ρ(A)vw T . First, if (ci, vi) is
an eigenpair of A − ρ(A)vw T and ci = 0, then (ci, vi) is an eigenpair of A.
We can see this by multiplying both sides of the eigen-equation by vw T :
hence,
Next, we show that
civw T vi = vw T A − ρ(A)vw T vi
= vw T A − ρ(A)vw T vw T vi
= ρ(A)vw T − ρ(A)vw T vi
= 0;
Avi = A − ρ(A)vw T vi
= civi.
Theory of Statistics c○2000–2013 James E. Gentle
ρ A − ρ(A)vw T < ρ(A). (0.3.71)