Question

Find the​ steady-state vector for the matrix below: {0.6, 0.3, 0.1}, {0, 0.2, 0.4}, {0.4, 0.5, 0.5}

The numbers listed here are the rows of a 3x3 matrix. Any help is appreciated as I do not understand steady state vectors very well

Answer 1

The given matrix is A =

0.6	0.3	0.1
0	0.2	0.4
0.4	0.5	0.5

If X = (x,y,z)^T is the steady state vector, then AX = X or, (A-I₃)X = 0. To solve this equation, we need to reduce the matrix A-I₃ to its RREF which is

1	0	-5/8
0	1	-1/2
0	0	0

Hence, the equation (A-I₃)X = 0 is equivalent to x-5z/8 = 0 or, x = 5z/8 and y-z/2 = 0 or, y = z/2. Then, X =(5z/8,z/2,z)^T =(z/8)(5,4,8)^T.

Hence, (5,4,8)^T is the steady state vector for the given matrix A.