Find a 2 A 2 stochastic matrix A that is not the identity th

Find a 2 A 2 stochastic matrix A, that is not the identity, that has [21/29 8/29] for a steady-state vector.

Solution

If A is a 2x2 stochastic matrix, then the entries of each column vectors are positive and their sum is 1. Further steady state vector x is given by Ax = x or, (A-I₂) x = 0. Let A =

a

b

1-a

1-b

Then A –I₂ =

a-1

b

1-a

-b

Since x = (21/29, 8/29)^T , we have (21/29)(a-1) +(8/29)b= 0 , or, on multiplying both the sides by 29, we have 21(a-1) + 8b = 0 or, 21a +8b = 21 …(1) and (21/29)(1-a) - (8/29)b = 0 , or, on multiplying both the sides by 29, we have, 21(1-a) - 8b = 0 or, 21a +8b =21 . ..(2). The two equations are identical and b = 21/8(1-a). Therefore, A =

a

21/8(1-a)

1-a

(21a-13)/8

Where a is an arbitrary positive real number.less than 1.

a	b
1-a	1-b