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Chapter 9: Problem 12

Find the steady-state vector for the transition matrix. $$ \left[\begin{array}{ll} .9 & 1 \\ .1 & 0 \end{array}\right] $$

Short Answer

Expert verified
The steady-state vector for the given transition matrix is approximately \(v \approx \begin{bmatrix} 0.9091 \\ 0.0909 \end{bmatrix}\).

Step by step solution

01

Set up the system of linear equations for the steady-state vector

Let the steady-state vector as \(v = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix}\). The matrix equation representing the steady-state condition is: $$ P \times v = v $$ Where \(P\) is the given transition matrix $$P = \begin{bmatrix} 0.9 & 1 \\ 0.1 & 0 \end{bmatrix}$$ Now substitute P and v in the matrix equation and we get: $$ \begin{bmatrix} 0.9 & 1 \\ 0.1 & 0 \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} $$
02

Solve the linear system of equations

Now we need to solve the linear system of equations. Multiply the matrices P and v: $$ \begin{bmatrix} 0.9 v_1 + v_2 \\ 0.1 v_1 \end{bmatrix} = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} $$ This matrix equation gives us two linear equations: 1. \(0.9 v_1 + v_2 = v_1\) 2. \(0.1 v_1 = v_2\) The first equation can be simplified as: $$ v_2 = 0.1 v_1 $$ So, from the second equation, we can directly say that \(v_2 = 0.1 v_1\). Also, note that \(v_1 + v_2 = 1\) since they are probabilities, and their sum should equal to 1.
03

Normalize the solution

Now, substitute the value of \(v_2\) into the normalization equation: $$ v_1 + 0.1 v_1 = 1 $$ Solve for \(v_1\) : $$ v_1 = \frac{1}{1.1} \approx 0.9091 $$ Now, calculate the value of \(v_2\) using the relation: $$ v_2 = 0.1 v_1 \approx 0.0909 $$ So, the steady-state vector v is: $$ v = \begin{bmatrix} 0.9091 \\ 0.0909 \end{bmatrix} $$ Thus, the steady-state vector for the given transition matrix is approximately \(\begin{bmatrix} 0.9091 \\ 0.0909 \end{bmatrix}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Transition Matrix
A transition matrix represents the probabilities of transitioning from one state to another in a stochastic process.

A common application is in Markov chains, where every entry in the matrix corresponds to the probability of moving from one state to another state. The matrix is square, meaning it has the same number of rows and columns, which represent the states of the system.

In our exercise, the transition matrix is:\[\begin{equation}\begin{bmatrix}0.9 & 1 \0.1 & 0\end{bmatrix}\end{equation}\]
This means that there is a 90% chance to stay in state 1, and a 10% chance of transitioning from state 1 to state 2. Conversely, if we are in state 2, we will certainly transition to state 1 in the next step.
System of Linear Equations
The system of linear equations arises when we look for a steady-state vector.

This is because we are looking for a vector that will not change when the transition matrix is applied to it, which mathematically is represented by the equation \[\begin{equation}P \times v = v\end{equation}\]
where P is the transition matrix and v is the steady-state vector we are trying to find. Our goal is to find values for the elements of vector v that satisfy this condition, which leads us to a system of equations.

Through matrix multiplication and comparison of both sides of the equation, we obtain our system of equations. The beauty of this process is that it can be used for any Markov process regardless of the size of the transition matrix, as long as the matrix is square.
Matrix Multiplication
The process of matrix multiplication plays a crucial role in finding the steady-state vector for a transition matrix.

When you multiply the transition matrix by the steady-state vector, you effectively conduct a series of multiplications and additions across corresponding elements in the rows and columns. For example:\[\begin{equation}\begin{bmatrix}a & b \c & d\end{bmatrix}\times\begin{bmatrix}x \y\end{bmatrix}= \begin{bmatrix}a\cdot x + b\cdot y \c\cdot x + d\cdot y\end{bmatrix}\end{equation}\]
By performing the matrix multiplication, we set up our system of equations, which is then solved using algebraic techniques or methods like substitution or elimination to find the solutions that satisfy the steady-state condition, P times v equals v. This process provides us with the values that are probabilities in the context of our problem.

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Most popular questions from this chapter

Determine the maximin and minimax strategies for each two-person, zero-sum matrix game. $$ \left[\begin{array}{rr} -1 & 3 \\ 2 & 5 \end{array}\right] $$

Determine whether the matrix is an absorbing stochastic matrix. \(\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\)

The registrar of Computronics Institute has compiled the following statistics on the progress of the school's students in their 2 -yr computer programming course leading to an associate degree: Of beginning students in a particular year, \(75 \%\) successfully complete their first year of study and move on to the second year, whereas \(25 \%\) drop out of the program; of second-year students in a particular year, \(90 \%\) go on to graduate at the end of the year, whereas \(10 \%\) drop out of the program. a. Construct the transition matrix associated with this Markov process. b. Compute the steady-state matrix. c. Determine the probability that a beginning student enrolled in the program will complete the course successfully.

Find the expected payoff \(E\) of each game whose payoff matrix and strategies \(P\) and \(Q\) (for the row and column players, respectively) are given. \(\left[\begin{array}{rr}3 & 1 \\ -4 & 2\end{array}\right], P=\left[\begin{array}{ll}\frac{1}{2} & \frac{1}{2}\end{array}\right], Q=\left[\begin{array}{l}\frac{3}{5} \\ \frac{2}{5}\end{array}\right]\)

Brady's, a conventional department store, and ValueMart, a discount department store, are each considering opening new stores at one of two possible sites: the Civic Center and North Shore Plaza. The strategies available to the management of each store are given in the following payoff matrix, where each entry represents the amounts (in hundreds of thousands of dollars) either gained or lost by one business from or to the other as a result of the sites selected. a. Show that the game is strictly determined. b. What is the value of the game? c. Determine the best strategy for the management of each store (that is, determine the ideal locations for each store).
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