The initial probability vector P and the transition matrix A are given for a three-state Markov chain: P = 0.1 0.4 0.5 A = 0.6 0 0.4 0.2 0.8 0.2 0.2 0.2 0.4 It is given that the eigenvalues of A are 1, 0.6, 0.2 and that all entries of the matrix A² are positive. Find the limit LP = lim Ak P. k→∞
The initial probability vector P and the transition matrix A are given for a three-state Markov chain: P = 0.1 0.4 0.5 A = 0.6 0 0.4 0.2 0.8 0.2 0.2 0.2 0.4 It is given that the eigenvalues of A are 1, 0.6, 0.2 and that all entries of the matrix A² are positive. Find the limit LP = lim Ak P. k→∞
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![The initial probability vector \( P \) and the transition matrix \( A \) are given for a three-state Markov chain:
\[ P = \begin{pmatrix} 0.1 \\ 0.4 \\ 0.5 \end{pmatrix} \]
\[ A = \begin{pmatrix} 0.6 & 0 & 0.4 \\ 0.2 & 0.8 & 0.2 \\ 0.2 & 0.2 & 0.4 \end{pmatrix} \]
It is given that the eigenvalues of \( A \) are 1, 0.6, 0.2 and that all entries of the matrix \( A^2 \) are positive. Find the limit \( LP = \lim_{k \to \infty} A^k P \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F32f77ee0-291c-46d0-b315-80fb2fd096d8%2F215cf4de-8a8c-4f85-b795-654949ae35c5%2Fn1va3m3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The initial probability vector \( P \) and the transition matrix \( A \) are given for a three-state Markov chain:
\[ P = \begin{pmatrix} 0.1 \\ 0.4 \\ 0.5 \end{pmatrix} \]
\[ A = \begin{pmatrix} 0.6 & 0 & 0.4 \\ 0.2 & 0.8 & 0.2 \\ 0.2 & 0.2 & 0.4 \end{pmatrix} \]
It is given that the eigenvalues of \( A \) are 1, 0.6, 0.2 and that all entries of the matrix \( A^2 \) are positive. Find the limit \( LP = \lim_{k \to \infty} A^k P \).
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