(e) If o0 < B _< P and p is an eigenvalue of B, then | p| r = 1. Moreover,
PI = r = 1 implies B= P.
(f) r = 1 is a simple root of the characteristic polynomial of P.
This theorem follows immediately from Theorem B2 by application of Corollary
B1 with T a stochastic primitive matrix, P. Among its consequences are the following.
Proposition B1: The right eigenvector asserted in Theorem B4(b) and (d) can be selected
as the vector 1.
This result follows from the row sum constraint on nxn stochastic matrices, which
can be expressed as Pi = 1. Thus, 1 is a right eigenvector, corresponding to eigenvalue 1,
of every nxn stochastic matrix. Theorem B4 asserts that for finite primitive stochastic
matrices, it is unique to within a nonzero scalar multiple.
Proposition B2: Let the vector q be the left eigenvector asserted in Theorem B4(b) and
(d). Then, the additional constraint q 1 = 1 is consistent and makes q unique.
Since the left eigenvector asserted in Theorem B4(b) and (d) has strictly positive
components, its inner product with the vector I is a strictly positive (nonzero) number.
Consequently, that inner product can be used to normalize the eigenvector to produce a q
which satisfies both requirements, and Proposition B2 follows.
Proposition B3: If P is an n x n stochastic primitive matrix, then
lim (-P)=q
k --.-
where q is the unique vector asserted in Proposition B2.