Find the steady-state vector associated with the given transition matrix. (Give exact answers. Do not round.) 0.2 0.8 0.1 0.9
Find the steady-state vector associated with the given transition matrix. (Give exact answers. Do not round.) 0.2 0.8 0.1 0.9
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Title: Understanding Steady-State Vectors in Transition Matrices**
---
**Find the steady-state vector associated with the given transition matrix. (Give exact answers. Do not round.)**
\[
\begin{bmatrix}
0.2 & 0.8 \\
0.1 & 0.9 \\
\end{bmatrix}
\]
This matrix represents a transition matrix often used in Markov processes. Each element in the matrix represents the probability of transitioning from one state to another. For example, in this matrix:
- 0.2 is the probability of staying in the first state.
- 0.8 is the probability of transitioning from the first state to the second state.
- 0.1 is the probability of transitioning from the second state to the first state.
- 0.9 is the probability of staying in the second state.
The goal is to find the steady-state vector, a vector that remains unchanged after applications of the transition matrix. This vector shows the long-term behavior of the system.
Below is a layout representing an unanswered system of equations, depicted by blank brackets, to compute the steady-state vector:
\[
\begin{bmatrix}
\text{\_\_\_} \\
\text{\_\_\_} \\
\end{bmatrix}
\]
Arrows indicate the direction of the equations, relating to inputs and transformations within the system.
**Key Concepts:**
1. **Transition Matrix**: A square matrix used to describe the transitions of a Markov chain. Each of its rows is a probability vector, meaning all elements are non-negative and each row sums to one.
2. **Steady-State Vector**: A probability vector, denoted usually by \(\pi\), that becomes stable, satisfying the condition \(\pi = \pi P\), where \(P\) is the transition matrix.
Understanding these concepts allows students to explore how systems evolve over time and reach equilibrium, which is integral in fields such as economics, genetics, and computer science.
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8533e211-c3b0-4c2c-a0a5-c3ff9e21dbc2%2Ff7be10ae-fbdf-4f9a-8150-d0d9066b2809%2Fu8mz7b_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Understanding Steady-State Vectors in Transition Matrices**
---
**Find the steady-state vector associated with the given transition matrix. (Give exact answers. Do not round.)**
\[
\begin{bmatrix}
0.2 & 0.8 \\
0.1 & 0.9 \\
\end{bmatrix}
\]
This matrix represents a transition matrix often used in Markov processes. Each element in the matrix represents the probability of transitioning from one state to another. For example, in this matrix:
- 0.2 is the probability of staying in the first state.
- 0.8 is the probability of transitioning from the first state to the second state.
- 0.1 is the probability of transitioning from the second state to the first state.
- 0.9 is the probability of staying in the second state.
The goal is to find the steady-state vector, a vector that remains unchanged after applications of the transition matrix. This vector shows the long-term behavior of the system.
Below is a layout representing an unanswered system of equations, depicted by blank brackets, to compute the steady-state vector:
\[
\begin{bmatrix}
\text{\_\_\_} \\
\text{\_\_\_} \\
\end{bmatrix}
\]
Arrows indicate the direction of the equations, relating to inputs and transformations within the system.
**Key Concepts:**
1. **Transition Matrix**: A square matrix used to describe the transitions of a Markov chain. Each of its rows is a probability vector, meaning all elements are non-negative and each row sums to one.
2. **Steady-State Vector**: A probability vector, denoted usually by \(\pi\), that becomes stable, satisfying the condition \(\pi = \pi P\), where \(P\) is the transition matrix.
Understanding these concepts allows students to explore how systems evolve over time and reach equilibrium, which is integral in fields such as economics, genetics, and computer science.
---
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
