Proposition 6.2: If the constraint described in Eq. 6.2 is used to replace any column (e.g.
column n) of the system in Eq. 6.1, the resulting system (Eq. 6.3) is full rank, or equiva-
lently, I (P- I),| 0.
Since P is a stochastic matrix (Definition A3), the system of equations in Eq 6.1 can
be transformed into an equivalent system in which the column indexed by the arbitrary
column index n e S' is represented by the equation O = 0. The required transformation
is obtainable by replacing column n by the sum of all columns m e S', and thus any
n e S' is a candidate for replacement. Proposition 6.2 is then a restatement of Proposition
B2 in terms of the determinant of the matrix of the modified system. It is the essential
condition for justification of the following proposition.
Proposition 6.3: The components of the stationary distribution can be expressed in the
form
(P I I
(M-
I ( (P- l1
where (P-I,) is derived from (P-I)n by replacing the row of (P-I)n indexed by m e S'
with the row vector e.
This result is simply an application of Cramer's Rule to the solution of the system
in Eq. 6.3. It applies because I (P I)| 0 is assured by Proposition 6.2.
The equality defined in Proposition 6.3 can be evaluated without computing
I (P- I)| directly, as suggested by the following proposition.
Proposition 6.4: The denominator determinant in Proposition 6.3 can be written as
P- -m S)
im C S'