Explanation Given information: The stochastic matrix P = [ 0 0 0.2 0.5 0.9 0 0.5 0.1 0.8 ] . Calculation: To find the steady state matrix of a Markov chain, use the following steps: 1) First check whether the matrix of transition probabilities P is a regular matrix. 2) To find the steady state matrix of a Markov chain X ¯ , solve the system of linear equations obtained from the matrix equation P X ¯ = X ¯ with the equation x 1 + x 2 + . . . + x n = 1 . 3) Then check the solution obtained in step 2 in the matrix equation P X ¯ = X ¯ . If some power of P has only positive entries, then the stochastic matrix P is regular. Consider the stochastic matrix P = [ 0 0 0.2 0.5 0.9 0 0.5 0.1 0.8 ] . P 2 = [ 0 0 0.2 0.5 0.9 0 0.5 0.1 0.8 ] [ 0 0 0.2 0.5 0.9 0 0.5 0.1 0.8 ] = [ 0.1 0.02 0.16 0.45 0.81 0.1 0.45 0.17 0.74 ] Second power of P , that is, P 2 has only positive entries. Hence, the stochastic matrix P = [ 0 0 0.2 0.5 0.9 0 0.5 0.1 0.8 ] is regular. Therefore, the matrix of transition probabilities P is regular. To find the steady state matrix X ¯ = [ x 1 x 2 x 3 ] of a Markov chain, Use the matrix equation P X ¯ = X ¯ . By substituting values of P and X ¯ , it gives [ 0 0 0.2 0.5 0.9 0 0.5 0.1 0.8 ] [ x 1 x 2 x 3 ] = [ x 1 x 2 x 3 ] By solving, 0.2 x 3 = x 1 ⇒ 0.2 x 3 − x 1 = 0 0.5 x 1 + 0.9 x 2 = x 2 ⇒ 0.5 x 1 − 0.1 x 2 = 0 0.5 x 1 + 0.1 x 2 + 0.8 x 3 = x 3 ⇒ 0.5 x 1 + 0.1 x 2 + 0.8 x 3 − x 3 = 0 ⇒ 0.5 x 1 + 0.1 x 2 − 0.2 x 3 = 0 Since x 1 + x 2 + x 3 = 1 Here, the following system of linear equations is obtained: 0.2 x 3 − x 1 = 0 --------- (1) 0.5 x 1 − 0.1 x 2 = 0 ---------- (2) 0.5 x 1 + 0.1 x 2 − 0

Elementary Linear Algebra - Text Only (Looseleaf)

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Chapter 2.CR, Problem 68CR

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Whether the stochastic matrix P is regular and then to find steady state matrix X¯ of the Markov chain with matrix of transition probabilities P where P=[000.20.50.900.50.10.8]

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Chapter 2.CR, Problem 67CR

Chapter 2.CR, Problem 69CR

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Whether the stochastic matrix P is regular and then to find steady state matrix X ¯ of the Markov chain with matrix of transition probabilities P where P = [ 0 0 0.2 0.5 0.9 0 0.5 0.1 0.8 ]

Solution Summary: The author explains how to determine whether the stochastic matrix P is regular and the steady state matrix of a Markov chain.

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Whether the stochastic matrix P is regular and then to find steady state matrix X¯ of the Markov chain with matrix of transition probabilities P where P=[000.20.50.900.50.10.8]

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