## Info

gives the sum of individuals initially in region I (0.8 x 5) plus the number of individuals initially in region 2 who move to region I (0.15 >: 10) plus the number of individuals initially in region 3 who move to region I (0.05 x 6). giving the sum 5.8. We can see that the population distribution between the three regions changed from x" to x1. with regions I and 3 gaining people, while region 2 is losing them. ■

### Definition 8.9

A special case of matrix multiplication is when a square matrix is raised to a power n. that is. the matrix is multiplied by itself n times. As with the case of ordinary algebra we give the following definition.

The matrix A" is the product matrix obtained by multiplying the square matrix A by itself n times.

### Regional Migration over Time

In example 8.11 we saw how the distribution of the populations of three regions in a country changes between two periods. The matrix equation that we used to obtain the distribution of these regional populations after one period, x1, is given by x' = Px", where x° is the distribution of the populations in the regions at time 0 and P is the transition matrix.

To determine how this distribution changes over n periods we then solve the following equation:

where x" describes the distribution of the regional populations ai lime period n and x"'1 the distribution at the previous period — I. However, from the equation above it becomes clear that x" 1 = Px" 2. Therefore, by substituting back into equation (8.1). we obtain x" - P:x" \ In fact, backward substitution to the vector of initial population distributions x" yields x" - P"\° (8.2)

Equation (8.2) describes the evolution of the populations between the regions after n periods. It is based on the product matrix P". We can actually think of k" = P"x" as a sequence of vectors in an analogous way to the sequences studied n chapter?, In chapter 10 we will study the behavior of such an equation in greater detail.

### Example 8.12 Regional Migration over Two Periods

Consider the transition matrix. P. and initial distribution of population x° given n example 8.11 Find the distribution of the population after two periods.

Solution