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{SECT 0 {PARA 260 "" 0 "" {TEXT -1 30 "Transition Probablity Matrices
" }}{PARA 0 "" 0 "" {TEXT -1 120 "Example: Please refer to the rental \+
car problem that is posted on 'http://www.radford.edu/~wyang/525/marko
v/markov.htm'." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Wa
rning, the protected names norm and trace have been redefined and unpr
otected\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "P := matrix([[0.8,0.3,0.2],[.1,.2,.
6],[0.1,0.5,0.2]]);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG-%'matr
ixG6#7%7%$\"\")!\"\"$\"\"$F,$\"\"#F,7%$\"\"\"F,F/$\"\"'F,7%F2$\"\"&F,F
/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "If  x0=[0,1,0] Let's see wha
t happens to (P^n)*x0  when n is large;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "x0:=transpose([[0,1,0]]);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G-%'matrixG6#
7%7#\"\"!7#\"\"\"F)" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 24 "x20:=evalm((P^20)&*x0);\n" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$x20G-%'matrixG6#7%7#$\"+s7utb!#57#$\"+&R(4&H#F,
7#$\"+L8;J@F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "x21:=evalm
((P^21)&*x0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$x21G-%'matrixG6#7%
7#$\"+.Xvtb!#57#$\"+2/4&H#F,7#$\"+\"4b68#F," }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 60 "Notice that x20 and x21 are the same up to 5 decimal pl
aces." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Let's see Pq =q has a so
lution, or if P has a steady state vector." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 22 "B:=evalm(P-diag(1$3));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"BG-%'matrixG6#7%7%$!\"#!\"\"$\"\"$F,$\"\"#F,7%$\"\"
\"F,$!\")F,$\"\"'F,7%F2$\"\"&F,F4" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "rref(B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG
6#7%7%\"\"\"\"\"!$!+;YQ:E!\"*7%F)F($!+xI#p2\"F,7%F)F)$F)F)" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 177 "This means that if P is given, we have i
nfinitely many solutions [x1, x2, x3]. If we pick x3 to be '0.21311613
33' then we should get a vector that is close to either x20 or x21." }
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