Given a situation in which 20% of the sane population becomes demented and 10 percent of the demented population becomes sane in a given transition period, a population initially consisting of 500 sane and 500 demented individuals will behave as indicated below.

• The middle picture in the figure below depicts the "flow" of 100 sane individuals to the demented side and 50 demented individuals to the sane side.
• We see that there is a net change of 50 individuals from sane to demented; i.e., the sane population will lose 50 individuals and the demented will gain 50.
• In the next transition we see that the net "flow" will be 35 individuals from the sane to the demented side.

The figure below depicts where the 450 sane and 550 demented people come from during the first transition.

• The sane population will be made up of the .80 * 500 = 400 who remained sane and the .10 * 500 = 50 demented individuals who become sane.
• The demented population will be made up of the .90 * 500 demented to remain demented and the .20 * 500 = 100 sane individuals who become demented.

The figure below depicts the multiplication by the resulting transition matrix of a population vector consisting of 600 sane and 400 demented.

• When we multiply the first row by the column vector the numbers we obtain represent the 80 percent of the sane who remain sane and the 10 percent of the demented who become sane, so that the first number we get in the new population vector must depict the number of sane individuals after the transition.
• When multiply the second row by the column vector the numbers we obtain represent the 20 percent of the sane who become demented and the 90 percent of the demented who remain demented, so that the second number in the new population vector must depict the number of demented individuals after the transition.

We could iterate this process by multiplying the new population vector [520 480 ]` by the transition matrix to obtain the population configuration for the following year.

• Since the [520 480 ]` was obtained by multiplying the original population vector [600 400]` by the transition matrix, we could rewrite the product at the top of the figure as in the middle of the figure.
• If we then group the two matrices on the left so that they are multiplied first (i.e., if we assume the an associate law for matrix multiplication), as at the bottom of the figure, we should obtain the same result.

Video File #01

To multiply the two matrices, we multiply each column of the second by each row of the first and place the results in the corresponding rows and columns of the product matrix.

• The result of multiplying the first row by the first column goes into the first row and first column of the product matrix.
• The result of multiplying the first row by the second column goes into the first row and second column of the product matrix.
• The result of multiplying the second row by the first column goes into the second row and first column of the product matrix.
• The result of multiplying the second row by the second column goes into the second row and second column of the product matrix.

Had we multiplied the population vector [520 480]` by the transition matrix we would have obtained [464 536]` for the end of the second year.

• Had we on the other hand multiplied the original population vector [600 400]` by the result [ [.66, .17], [.34, .83] ] of the preceding matrix calculation we would have obtained the same result without having had to compute the intermediate population configuration [520 480 ]`.

Video File #02

The matrix [ [.66, .17], [.34, .83] ] was obtained by multiplying the original transition matrix [ [.8, .1], [.2, .9]] by itself, and so is seen to be the square of that matrix.

• We thus see that multiplying by the square of a transition matrix gives is a 2-transition matrix, i.e., the matrix we multiply by an original population vector to get the population after two transitions.
• If we let A stand for the original transition matrix, the new matrix is A² = [ [.66, .17], [.34, .83] ]

Video File #03

We also note that A¹⁰⁰ = [ [.333333, .333333], [.666666, .666666] ].

• You should check that multiplying A¹⁰⁰ by [ 500, 500]` yields [333.3..., 666.6...]`, which is the population configuration we expect after a long time.
• You should also check that multiplying A¹⁰⁰ by [1000, 0]` or by [0, 1000]`yields the same result, which should make it very plausible to you that the population configuration will after a time approach the same value regardless of the initial population configuration.

We note that A⁵⁰ is also equal to [ [.333333, .333333], [.666666, .666666] ], but that A³⁰ is equal to these values only to a few decimal places.

• We expect that A¹⁰⁰ will be equal to the limiting matrix [ [1/3, 1/3], [2/3, 2/3] ] to a greater number of decimal places than A^50.

A graph of population D vs. population S is depicted below.

• Each population might be represented by a population vector consisting of an arrow from the origin to the graph point corresponding to that population.
• While the figure is not drawn with great accuracy, we see that since the changes in the population configuration become less and less with each iteration of the process, the vectors will tend to get closer and closer together as they approach the limiting population vector (333.3..., 666.6...).

The answer is that the points lie on a straight line.

• This follows from the fact that the total number of people is 1000 so that S + D = 1000.
• It follows that D = 1000 - S.
• In y vs. x notation, we have y = 1000 - x, which is a straight line with vertical intercept 1000 and slope -1.