Diagonalise to solve Markov process 4/4/2002 Math 21b, O. KnillThe percentage of people using Apple OS X or the Gnu/Linuxoperating system is represented by a vector ~v =ml. Let 2/3be the percentage of Mac OS X users, who switch to Linux eachmonth and 1/2 the percentage of Linux OS users, who switch toApple each month. What is the distribution Linux/OSX usersafter 2 years if initially both groups have the same number ofusers.The matrix A =1/3 1/22/3 1/2. has the property that Amlis the distribution a month later. It is called astochastic matrix: the sum in each column is 1. The dynamics x(t) 7→ Ax(t) = x(t + 1) is called a Markovchain.AIM: COMPUTE Ak1/21/2, the distribution after k months for k = 1 and if you want for k = 2.COMPUTE THE EIGENVALUES OF A.COMPUTE THE EIGENVECTORS OF A.WHAT IS THE MATRIX S for which S−1AS = B is diagonal.WHAT IS B24, up to 10 digits accuracy?WHAT IS A24?WHAT IS THE DISTRIBUTION AFTER 24
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