A First Course in Linear Algebra, 2017a

7.3. Applications of Spectral Theory 375
=
=
⎡
⎣
⎡
⎣
0.4 0.1 0.5
0.4 0.6 0.1
0.2 0.3 0.4
0.4
0.4
0.2
⎤
⎦
⎤⎡
⎦⎣
1
0
0
⎤
⎦
X 2 = AX
⎡ 1
⎤⎡
0.4 0.1 0.5
= ⎣ 0.4 0.6 0.1 ⎦⎣
0.2 0.3 0.4
⎡ ⎤
0.3
= ⎣ 0.42 ⎦
0.28
0.4
0.4
0.2
⎤
⎦
This gives the probabilities that our walker ends up in locations 1, 2, and 3. For this example we are
interested in location 3, with a probability on 0.28.
♠
Returning to the context of migration, suppose we wish to know how many residents will be in a
certain location after a very long time. It turns out that if some power of the migration matrix has all
positive entries, then there is a vector X s such that A n X 0 approaches X s as n becomes very large. Hence as
more time passes and n increases, A n X 0 will become closer to the vector X s .
Consider Theorem 7.33. Letn increase so that X n approaches X s .AsX n becomes closer to X s ,sotoo
does X n+1 . For sufficiently large n, the statement X n+1 = AX n can be written as X s = AX s .
This discussion motivates the following theorem.
Theorem 7.36: Steady State Vector
Let A be a migration matrix. Then there exists a steady state vector written X s such that
X s = AX s
where X s has positive entries which have the same sum as the entries of X 0 .
As n increases, the state vectors X n will approach X s .
Note that the condition in Theorem 7.36 can be written as (I − A)X s = 0, representing a homogeneous
system of equations.
Consider the following example. Notice that it is the same example as the Example 7.34 but here it
will involve a longer time frame.