A First Course in Linear Algebra, 2017a

7.3. Applications of Spectral Theory 373
find the population of location i after k years, we compute the i th component of A k X. This discussion is
summarized in the following theorem.
Theorem 7.33: State Vector
Let A be the migration matrix of a population and let X n be the vector whose entries give the
population of each location at time period n. ThenX n is the state vector at period n and it follows
that
X n+1 = AX n
The sum of the entries of X n will equal the sum of the entries of the initial vector X 0 . Since the columns
of A sum to 1, this sum is preserved for every multiplication by A as demonstrated below.
∑
i
Consider the following example.
∑
j
a ij x j = ∑
j
Example 7.34: Using a Migration Matrix
Consider the migration matrix
⎡
A = ⎣
x j
(∑
i
.6 0 .1
.2 .8 0
.2 .2 .9
)
a ij = ∑x j
j
for locations 1,2, and 3. Suppose initially there are 100 residents in location 1, 200 in location 2 and
400 in location 3. Find the population in the three locations after 1,2, and 10 units of time.
⎤
⎦
Solution. Using Theorem 7.33 we can find the population in each location using the equation X n+1 = AX n .
For the population after 1 unit, we calculate X 1 = AX 0 as follows.
⎡
⎣
X 1 = AX
⎡ 0
⎤
x 11
x 21
⎦ =
x 31
=
⎣
⎡
⎣
.6 0 .1
.2 .8 0
.2 .2 .9
100
180
420
Therefore after one time period, location 1 has 100 residents, location 2 has 180, and location 3 has 420.
Notice that the total population is unchanged, it simply migrates within the given locations. We find the
locations after two time periods in the same way.
⎡
⎣
X 2 = AX
⎡ 1
⎤
x 12
x 22
⎦ =
x 32
⎣
⎤
⎦
.6 0 .1
.2 .8 0
.2 .2 .9
⎤⎡
⎦⎣
⎤⎡
⎦⎣
100
200
400
100
180
420
⎤
⎦
⎤
⎦