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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