Find the stable vector of 1 3 4 4 P = 4 %3D 3 7
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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![**Problem Statement:**
Find the stable vector of the matrix \( P \).
\[ P = \begin{bmatrix} \frac{1}{4} & \frac{3}{4} \\ \frac{4}{7} & \frac{3}{7} \end{bmatrix} \]
**Explanation:**
Given a transition matrix \( P \), you are tasked to find its stable vector. This involves identifying the vector that remains unchanged when \( P \) is applied repeatedly, typically corresponding to the steady state of a Markov process.
To solve for the stable vector \( \mathbf{v} \), satisfy the condition:
\[ P \mathbf{v} = \mathbf{v} \]
alongside the requirement that the components of \( \mathbf{v} \) sum to 1, since they represent probabilities.
The placeholders next to the matrix suggest input boxes for components of the stable vector you need to determine.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F73f3aba6-2965-4a12-ac09-7f2036eb7204%2F3eb754d8-a071-4261-ab69-e4e11ff603c2%2F7rvvzx_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the stable vector of the matrix \( P \).
\[ P = \begin{bmatrix} \frac{1}{4} & \frac{3}{4} \\ \frac{4}{7} & \frac{3}{7} \end{bmatrix} \]
**Explanation:**
Given a transition matrix \( P \), you are tasked to find its stable vector. This involves identifying the vector that remains unchanged when \( P \) is applied repeatedly, typically corresponding to the steady state of a Markov process.
To solve for the stable vector \( \mathbf{v} \), satisfy the condition:
\[ P \mathbf{v} = \mathbf{v} \]
alongside the requirement that the components of \( \mathbf{v} \) sum to 1, since they represent probabilities.
The placeholders next to the matrix suggest input boxes for components of the stable vector you need to determine.
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