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Chapter 6: Problem 13

Show that when \(\mathbf{x}\) is a probability vector and \(A\) is a stochastic matrix, then \(A \mathbf{x}\) is another probability vector.

Short Answer

Expert verified
The product \(A \mathbf{x}\) is a probability vector because each entry of the resulting row vector is non-negative, as the product \(A_{ij} x_i\) is non-negative due to the definitions of a probability vector and a stochastic matrix. The sum of the components of the row vector \(A \mathbf{x}\) equals 1, as the inner sum \(\sum_{j=1}^n A_{ij}\) sums to 1 for each column of the stochastic matrix \(A\) and the outer sum \(\sum_{i=1}^n x_i\) sums to 1 since \(\mathbf{x}\) is a probability vector.

Step by step solution

01

Definitions

Recall the definitions of a probability vector and a stochastic matrix. - Probability vector, \(\mathbf{x}\): A row vector such that each element in the vector is non-negative (\(x_i \geq 0\)) and their sum equals 1 (\(\sum_{i=1}^n x_i = 1\)). - Stochastic matrix, \(A\): A square matrix such that each column is a probability vector.
02

Verify non-negativity

To show that the product \(A \mathbf{x}\) is non-negative, we must show that each entry in the resulting row vector is non-negative. The \((1, j)\)-entry of the product \(A \mathbf{x}\) is given by: \([(A \mathbf{x})]_{1j} = \sum_{i=1}^n A_{ij} x_i\) Since each \(x_i\) is non-negative (from the definition of a probability vector) and each \(A_{ij}\) is non-negative (from the definition of a stochastic matrix), the product \(A_{ij} x_i\) is non-negative as well. Then, the sum of non-negative terms is also non-negative, ensuring that the \((1, j)\)-entry of the resulting row vector \(A \mathbf{x}\) is non-negative.
03

Verify that the sum equals 1

Now, we must show that the sum of components of the row vector \(A \mathbf{x}\) equals 1. The sum of components of the row vector \(A \mathbf{x}\) is given by: \(\sum_{j=1}^n [(A \mathbf{x})]_{1j} = \sum_{j=1}^n \sum_{i=1}^n A_{ij} x_i = \sum_{i=1}^n x_i \sum_{j=1}^n A_{ij}\) Now, using the given definitions: - The inner sum \(\sum_{j=1}^n A_{ij}\) is equal to 1, as each column of the stochastic matrix \(A\) is a probability vector. - The outer sum \(\sum_{i=1}^n x_i\) is equal to 1, as \(\mathbf{x}\) is a probability vector. Thus, the sum of components of the row vector \(A \mathbf{x}\) evaluates to \(1 \cdot 1 = 1\).
04

Conclusion

We have demonstrated that each entry of the product \(A \mathbf{x}\) is non-negative, and the sum of its components equals 1. Therefore, \(A \mathbf{x}\) is a probability vector.

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Key Concepts

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

Probability Vector
A probability vector is a special kind of vector, often used in probability and statistics. This vector holds a collection of numbers, each representing a probability. For a vector to be a probability vector, it must meet two important criteria:
  • All entries must be non-negative, meaning that no entry can be less than zero. This is because probabilities cannot be negative.
  • The sum of all the entries in the vector must equal 1. This represents the idea that the total probability of all possible outcomes is 100% or 1.
Imagine a probability vector like a pie chart where each slice represents a probability. All slices together make a full pie, meaning their total is 1!
Non-Negativity
Non-negativity is a crucial concept when dealing with probability vectors and stochastic matrices. In these mathematical tools, non-negativity ensures that each element of the vector or matrix is zero or positive. Probabilities cannot be negative by their nature.
  • In a probability vector, each entry must be non-negative because it represents a probability.
  • For a stochastic matrix, a similar rule applies; each entry must be non-negative since it partially represents probabilities of transitions.
By ensuring non-negativity, we maintain sensible and valid probability values that can be used for real-world applications.
Matrix Multiplication
Matrix multiplication is a key operation in linear algebra, commonly used when working with vectors and matrices. The multiplication of a matrix by a vector involves taking the dot product of rows of the matrix with the vector. Here's how it works with a stochastic matrix and a probability vector:
  • We multiply each entry of the probability vector by the corresponding entry in each column of the matrix.
  • The results are added together to produce a single entry in a new row vector.
This process is repeated for each row until a new row vector is formed. When you multiply a stochastic matrix by a probability vector, the result is another probability vector, preserving the non-negativity and sum-to-one property.
Vector Operations
Vector operations include any mathematical computations that involve vectors, such as addition, subtraction, scaling, and multiplication. In our specific context involving probability vectors and stochastic matrices, these operations become slightly specialized.
  • Vector addition involves adding the corresponding components of two vectors together to form a new vector.
  • For probability vectors, while you can scale a vector by multiplying it with a scalar, the results must still need to sum to 1 to remain a probability vector.
  • Vector multiplication can refer to the dot product or, more relevant here, multiplying a vector by a matrix.
These operations allow us to transform and manipulate vectors to gain insights and solve problems in probabilistic models.

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Most popular questions from this chapter

a. Suppose \(A\) is a symmetric \(n \times n\) matrix. Using the Spectral Theorem, prove that if \(A \mathbf{x} \cdot \mathbf{x}=0\) for every vector \(\mathbf{x} \in \mathbb{R}^{n}\), then \(A=\mathrm{O}\). b. Give an example to show that the hypothesis of symmetry is needed in part \(a\).

Show that the product of the roots (real and complex) of the characteristic polynomial of \(A\) is equal to det \(A\). (Hint: If \(\lambda_{1}, \ldots, \lambda_{n}\) are the roots, show that \(p(t)=\) \(\pm\left(t-\lambda_{1}\right)\left(t-\lambda_{2}\right) \cdots\left(t-\lambda_{n}\right)\).)

Jane, Dick, and Spot are playing Frisbee. Dick is equally likely to throw to Jane or Spot; Jane always throws the Frisbee to Spot; and Spot is three times as likely to bring the Frisbee to Jane as to Dick. In the long run, what is the probability that Dick gets the Frisbee? (Be sure to check that the transition matrix here is regular.)

Suppose \(A\) and \(B\) are symmetric and \(A B=B A\). Prove there is an orthogonal matrix \(Q\) so that both \(Q^{-1} A Q\) and \(Q^{-1} B Q\) are diagonal. (Hint: Let \(\lambda\) be an eigenvalue of \(A\). Use the Spectral Theorem to show that there is an orthonormal basis for \(\mathbf{E}(\lambda)\) consisting of eigenvectors of \(B\).)

Each month U-Haul trucks are driven among the cities of Atlanta, St. Louis, and Poughkeepsie. \(1 / 2\) of the trucks in Atlanta remain there, while the remaining trucks are split evenly between St. Louis and Poughkeepsie. 1/3 of the trucks in St. Louis stay there, \(1 / 2\) go to Atlanta, and the remaining \(1 / 6\) venture to Poughkeepsie. And \(1 / 5\) of the trucks in Poughkeepsie remain there, \(1 / 5\) go to St. Louis, and 3/5 go to Atlanta. Show that the distribution of U-Haul trucks approaches a steady state, and find it.
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