2 MARKOV CHAINS 16
– Let now all eigenvalues θ
1
, . . . , θ
`
of A be pairwise different and let the eigenvectors φ
1
, . . . , φ
k−1
be linearly independent for a certain k ≤ ` .
– In order to show the independence of φ
1
, . . . , φ
k
it suffices to show that
k
X
j=1
a
j
φ
j
= 0 (32)
implies a
1
= . . . = a
k
= 0.
– Let a
1
, . . . , a
k
be such that (32) holds. This also implies
0 = A 0 =
k
X
j=1
a
j
Aφ
j
=
k
X
j=1
a
j
θ
j
φ
j
.
– The same argument yields
0 = θ
k
0 = θ
k
k
X
j=1
a
j
φ
j
=
k
X
j=1
θ
k
a
j
φ
j
and thus
0 =
k−1
X
j=1
(θ
k
− θ
j
)a
j
φ
j
.
– As the eigenvectors φ
1
, . . . , φ
k−1
are linearly independent
(θ
k
− θ
1
)a
1
= (θ
k
− θ
2
)a
2
= . . . = (θ
k
− θ
k−1
)a
k−1
= 0
and hence a
1
= a
2
= . . . = a
k−1
= 0 as θ
k
6= θ
j
for 1 ≤ j ≤ k − 1.
– Now (32) immediately implies a
k
= 0.
• If the eigenvalues θ
1
, . . . , θ
`
of A are pairwise distinct,
– the ` × ` matrix Φ consists of ` linearly independent column vectors,
– and thus Φ is invertible.
– Consequently, the matrix Ψ of the left eigenvectors is simply the inverse Ψ = Φ
−1
. This imme-
diately implies (31). ¤
2.2 Ergodicity and Stationarity
2.2.1 Basic Definitions and Quasi-positive Transition Matrices
• If the Markov chain X
0
, X
1
, . . . has a very large number ` of possible states, the spectral representation (30)
of the n-step transition matrix P
(n)
= P
n
discussed in Section 2.1.4 turns out to be inappropriate in order
to calculate
– the conditional probabilities p
(n)
ij
= P (X
n
= j | X
0
= i) of the random state X
n
– as well as the (unconditional) probabilities P (X
n
= j) =
P
`
i=1
α
i
p
(n)
ij
of X
n
after n À 1 (time-) steps.
• However, there are certain conditions
– ensuring the existence of the limits lim
n→∞
p
(n)
ij
and lim
n→∞
P (X
n
= j), respectively, as well as their
equality and independence of i,
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