# For the Transition matrix P= [ 0.5 0.5 ], solve the equation SP=S to find the stationary matrix S and...

For the Transition matrix P= [ 0.5 0.5 ], solve the equation SP=S to find the stationary matrix S and the limiting matrix P [ 0.6 0.4 ]

## answer

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x x xxxx xxxx

xxxx 0.4]

let S be xx xx

xxxx xxxx xxxxxxxx SP=S

we xxxx

xx y] xxxx xxxx x xx xx

[0.6 xxxx

xx> [(0.5x x xxxxx (0.5x x 0.4y)] = [x xx

by xxxxxxxxx both vectors xx xxxx

0.5x + 0.6y = x xxx xxxx x xxxx x x

xx > xxxx x xxxx and 0.5x = xxx

xxx we know xxxx x + x = 1

hence, take x = 1-y

and from xxxx = 0.5x

xx xxxx xxxx x 0.5(1 - xx

xx> xxxx = 0.5

xx> x = xxxxxxx = xxxx x 0.455

and xxxxxx x = xxx = xxxxxx x xxxxx

hence, xxxxxxxxxx xxxxxx S = xxxxxx 0.455]

xxxxxx P is xxxxxxxxx xxxxxx diagonal xxxxxxx are xxx zero} and irreducible xxxxxxx xxxxx are only xxx states and prob. xx xxxxx from one state xx another xx non xxxxx

Hence, for xxxxxxxxxxx and xxxxxxxxx xxxxxx limiting matrix xx given xx matrix with stationary xxxxx on xxxx xxxx

xxxxxx limiting xxxxxx xxx this P is given xx

Q = [0.545 0.455]

[0.545 0.455]

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