MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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Question
**Finding the Equilibrium Vector for a Transition Matrix** To determine the equilibrium vector for the given transition matrix: \[ \begin{bmatrix} \frac{1}{5} & \frac{4}{5} \\ \frac{1}{9} & \frac{8}{9} \end{bmatrix} \] *Steps to Find the Equilibrium Vector:* 1. **Understand the Matrix**: - The matrix is a \(2 \times 2\) transition matrix where each element represents the probability of transitioning from one state to another. 2. **Equilibrium Vector**: - The equilibrium vector, usually denoted as \(\vec{v}\), is a steady-state vector that satisfies \(\text{Matrix} \times \vec{v} = \vec{v}\). - Each element of the equilibrium vector represents a steady-state probability associated with each state. 3. **Solution Requirement**: - Solve the equation \(\begin{bmatrix} \frac{1}{5} & \frac{4}{5} \\ \frac{1}{9} & \frac{8}{9} \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x \\ y \end{bmatrix}\). - Additionally, ensure \(x + y = 1\) as it must form a probability distribution. 4. **Finding the Values**: - Calculate using algebraic or computational tools to derive the values of \(x\) and \(y\) that satisfy the above conditions. *Input & Instructions*: - Enter your solution in the provided box with simplified fractions or integers for each element of the equilibrium vector. **Note**: Carefully solve the system either by substitution or matrix algebra to confirm that the vector sum equals 1. (Typing an integer or simplified fraction for each matrix element is required in the provided box for the equilibrium vector.)
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Transcribed Image Text:**Finding the Equilibrium Vector for a Transition Matrix** To determine the equilibrium vector for the given transition matrix: \[ \begin{bmatrix} \frac{1}{5} & \frac{4}{5} \\ \frac{1}{9} & \frac{8}{9} \end{bmatrix} \] *Steps to Find the Equilibrium Vector:* 1. **Understand the Matrix**: - The matrix is a \(2 \times 2\) transition matrix where each element represents the probability of transitioning from one state to another. 2. **Equilibrium Vector**: - The equilibrium vector, usually denoted as \(\vec{v}\), is a steady-state vector that satisfies \(\text{Matrix} \times \vec{v} = \vec{v}\). - Each element of the equilibrium vector represents a steady-state probability associated with each state. 3. **Solution Requirement**: - Solve the equation \(\begin{bmatrix} \frac{1}{5} & \frac{4}{5} \\ \frac{1}{9} & \frac{8}{9} \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x \\ y \end{bmatrix}\). - Additionally, ensure \(x + y = 1\) as it must form a probability distribution. 4. **Finding the Values**: - Calculate using algebraic or computational tools to derive the values of \(x\) and \(y\) that satisfy the above conditions. *Input & Instructions*: - Enter your solution in the provided box with simplified fractions or integers for each element of the equilibrium vector. **Note**: Carefully solve the system either by substitution or matrix algebra to confirm that the vector sum equals 1. (Typing an integer or simplified fraction for each matrix element is required in the provided box for the equilibrium vector.)
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