Mathematics, 14.06.2020 05:57 Buxtoman8431
Exercise 7.3.5 The following is a Markov (migration) matrix for three locations 1 5 1 5 2 5 2 5 2 5 1 5 2 5 2 5 2 5 7.3. Applications of Spectral Theory 397 (a) Initially, there are 130 individuals in location 1, 300 in location 2, and 70 in location 3. How many are in each location after two time periods? (b) The total number of individuals in the migration process is 500. After a long time, how many are in each location?
Answers: 1
![\left[\begin{array}{ccc}140\\160\\200\end{array}\right]](/tpl/images/0685/7952/1adba.png)
![\bf M=\left[\begin{array}{ccc}\frac{1}{5}&\frac{1}{5}&\frac{2}{5}\\\frac{2}{5}&\frac{2}{5}&\frac{1}{5}\\\frac{2}{5}&\frac{2}{5}&\frac{2}{5}\end{array}\right]](/tpl/images/0685/7952/244ce.png)
![\bf P=\left[\begin{array}{ccc}130\\300\\70\end{array}\right]](/tpl/images/0685/7952/bf333.png)
![P_2=[P].[M]^2=[P].[M].[M]](/tpl/images/0685/7952/72f03.png)
![M^2=\left[\begin{array}{ccc}\frac{1}{5}&\frac{1}{5}&\frac{2}{5}\\\frac{2}{5}&\frac{2}{5}&\frac{1}{5}\\\frac{2}{5}&\frac{2}{5}&\frac{2}{5}\end{array}\right]\times \left[\begin{array}{ccc}\frac{1}{5}&\frac{1}{5}&\frac{2}{5}\\\frac{2}{5}&\frac{2}{5}&\frac{1}{5}\\\frac{2}{5}&\frac{2}{5}&\frac{2}{5}\end{array}\right]](/tpl/images/0685/7952/28953.png)
![=\frac{1}{25} \left[\begin{array}{ccc}7&7&7\\8&8&8\\10&10&10\end{array}\right]](/tpl/images/0685/7952/4dab6.png)
![\therefore\;\;\;\;\;\;\;\;\;\;\;P_2=\left[\begin{array}{ccc}7&7&7\\8&8&8\\10&10&10\end{array}\right] \times\left[\begin{array}{ccc}130\\300\\70\end{array}\right] = \left[\begin{array}{ccc}140\\160\\200\end{array}\right]](/tpl/images/0685/7952/bff01.png)
![After\;inifinite\;period=[M]^n.[P]](/tpl/images/0685/7952/5691e.png)
![Then,\;we\;get\;the\;same\;result\;if\;we\;measure [M]^n=\frac{1}{25} \left[\begin{array}{ccc}7&7&7\\8&8&8\\10&10&10\end{array}\right]](/tpl/images/0685/7952/8c6c9.png)
![=\left[\begin{array}{ccc}140\\160\\200\end{array}\right]](/tpl/images/0685/7952/a6a44.png)
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Exercise 7.3.5 The following is a Markov (migration) matrix for three locations 1 5 1...
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