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Perron's Formula expresses the elements
a_{i j}^{(n)}
of the nth power A^{n} of
a
square matrix A in terms of the eigenvalues and minor determinants of A.
An eigenvalue
with multiplicity
of A is a root
with multiplicity of the characteristic polynomial^{2.3}
,
where I_{m} is the
identity matrix. Denote the minor of a matrix A which arises
from deleting the ith row and jth column by A_{i j} and denote
the socalled minor of the determinant
which
equals
(1)^{i+j} times the determinant of the minor of
by
.
Note the correct order of i and j in the latter definition.
From [49, p. 16] we cite the following theorem:
Theorem 2.15 (Perron's Formula)
Let
be the eigenvalues of a
matrix
A with
multiplicities
,
respectively. Define
for every
a polynomial
of degree
by
.
Then for all
,
and for all
,
As we will see below,
Perron's formula is particularly useful if the matrix A has a maximal eigenvalue
with multiplicity 1 and if all other eigenvalues satisfy
.
This justifies the following definition and lemma which we cite from
[49, pp. 21 and 46].
Definition 2.10
A chain
is called regular if
has a maximal eigenvalue
with multiplicity 1 and all other eigenvalues satisfy
.
The chain is called positively regular if, in addition, all the numbers
P_{j} are strictly positive.
Lemma 2.16
A chain is regular (positively regular) if and only if there exists an integer
nsuch that for at least one
(for all
)
and for all
the probability
p^{(n)}_{i j}>0.
If a regular chain is irreducible, it is positively regular.
For a proof see [49, Theorem 14.I and 14.II].
The next corollary, which follows trivially from the definitions of irreducibility and
aperiodicity, gives an important class of positively regular chains.
By Corollary 2.17 the transition matrix
of an
irreducible aperiodic chain has an eigenvalue
with multiplicity 1and satisfies
for all
.
In this case, Perron's formula for the n'th power of
assumes the following form,
see [49, p. 19]:

(2.6) 
where

(2.7) 
Here
as
since the remaining term

(2.8) 
tends to zero on behalf of
,
,
see
[49, p. 21].
Since the irreducible aperiodic chain
has a finite state space
here, we already know from Lemma 2.8 that
for all pairs
of states
the n'th order transition probabilities
p^{(n)}_{i j}
tend to the probabilities P_{j} of the stable distribution
,
i.e. it holds
.
By Lemma 2.7, the stable distribution
is representable as
P_{i} = 1/E_{i}[T_{i}].
Thus we get the following two representations for the stable distribution
:

(2.9) 
Moreover, in [49, Chapter 3] it is shown that (2.7) also represents the stable
distribution in the case of a periodic irreducible chain although the remaining
term (2.8) will not tend to zero in this case due to the existence of eigenvalues
with
,
recall Theorem 2.14.
Next: Frequency Analysis
Up: Spectral Analysis of Markov
Previous: Spectral Analysis of Markov
Stefan Wegenkittl
19980519