Theorem B3: Let 7 be an nxn nonnegative primitive matrix with Perron-Frobenius eigen-
value r, let v and w be strictly positive left and right eigenvectors respectively of T
corresponding to r with v and w normed so that Vw = 1, and let the t I n distinct
eigenvalues of T be ordered such that r > | I2 31 3 *** > | >XJ with the additional condi-
tion that I I21 has multiplicity m2 equal to or greater than the multiplicity of any other
eigenvalue \ for which I|1 = I j. It follows that
(a) if X2 0, then as k oo
T = r + O(k'l 1 k)
elementwise, where s = m2- 1;
if 2 = 0, then for k 2 n 1
k = TWv
rV.
B.3 The Perron-Frobenius Theory for Stochastic Matrices
A stochastic matrix (e.g. the state transition matrix of a Markov chain) is a special
case of a square nonnegative matrix in which all row sums are equal to the constant 1.
The following results specialize those of Section B.2 to the case of T an nxn stochastic
primitive matrix, P.
Theorem B4: Let P be an nxn stochastic primitive matrix. Then
(a) r= 1 is an eigenvalue of P
(b) r = 1 has corresponding left and right eigenvectors with strictly positive
components
(c) r = 1 > I 1 for any eigenvalue X # r
(d) the eigenvectors associated with r= 1 are unique to constant multiples