where pO) is defined as before and where p) is defined as the matrix which results by
m m
replacing row m of P by the row vector e.
This result can be further reduced by noting that the row replacement by which PR
is generated from P preserves the row sum constraint (i.e. P- is a stochastic matrix).
Thus, I is an eigenvalue of P (Definition A3), from which it follows that I (Pj )| = 0.
Consequently, the m = n and m # n cases can be assembled as indicated by the following
proposition.
Proposition 6.6: The determinant I (P-I) ~ I defined in Proposition 6.5 can be written
as
(P I) F C=- -)
By collecting the results of Propositions 6.3-6 and noting that the superscript in P)
is now superfluous, the components of the stationary distribution can be written as indi-
cated in the following proposition.
Proposition 6.7: The components of the stationary distribution can be expressed in the
form
P 1| 1 PI II
nE S' ne S'
where Pm and Pg are derived from P by replacing the rows indexed by m and n respec-
tively with the row vector 6r.
Thus, computing the stationary distribution components reduces to evaluating the
characteristic polynomials of the Pn's at X = 1 (i.e. P(X) = I P- -II =I I P -1 = 1 (1)).
Also, since 1 is an eigenvalue of P it follows that ((1) = P II = 0, which suggests the
following alternative to Proposition 6.7. Its usefulness is established in Sections 9.3-9.4.