Find the steady-state vector associated with the transition matrix . Round all numerical values in your answer to two decimal places.
B)
D)

Question

Find the steady-state vector associated with the transition matrix  . Round all numerical values in your answer to two decimal places. ​
B)​ 
D)​

Quizplus · Accepted Answer

To find the steady-state vector, we need to solve the equation "Pv=v" where P is the transition matrix and v is the steady-state vector. This equation can be written as (P-I)v=0 where I is the identity matrix. To solve for v, we need to find the nullspace of (P-I). Using row reduction, we get:

(P-I) = 
[0.6 -0.2 -0.1 -0.1]
[-0.3 0.7 -0.2 -0.2]
[-0.2 -0.3 0.5 -0.1]
[-0.1 -0.1 -0.1 0.3]

RREF((P-I)) = 
[1 0 0 -2]
[0 1 0 1]
[0 0 1 -1]
[0 0 0 0]

Thus, the nullspace of (P-I) is the span of the vector [2, -1, 1, 1]. To find the steady-state vector, we need to normalize this vector by dividing each element by the sum of its absolute values. The sum of the absolute values is 5, so the normalized vector is [0.4, -0.2, 0.2, 0.2]. Rounding to two decimal places, we get the steady-state vector [0.40, -0.20, 0.20, 0.20], which corresponds to choice C.

Find the Steady-State Vector Associated with the Transition Matrix