An n n matrix A is said to be a stochastic matrix if all its entries are nonnegative and

Chapter 8, Problem 32

(choose chapter or problem)

An \(n\ \times\ n\) matrix A is said to be a stochastic matrix if all its entries are nonnegative and the sum of the entries in each row (or the sum of the entries in each column) add up to 1. Stochastic matrices are important in probability theory.

(a) Verify that

\(\mathbf{A}=\left(\begin{array}{ll} p & 1-p \\ q & 1-q \end{array}\right), \quad 0 \leq p \leq 1,0 \leq q \leq 1\),

and

\(\mathbf{A}=\left(\begin{array}{ccc} \frac{1}{2} & \frac{1}{4} & \frac{1}{4} \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ \frac{1}{6} & \frac{1}{3} & \frac{1}{2} \end{array}\right)\)

(b) Use a CAS or linear algebra software to find the eigenvalues and eigenvectors of the the \(3\ \times\ 3\) matrix A in part (a). Make up at least six more stochastic matrices of various sizes, \(2\ \times\ 2,\ 3\ \times\ 3,\ 4\ \times\ 4,\ \text{and}\ 5\ \times\ 5\). Find the eigenvalues and eigenvectors of each matrix. If you discern a pattern, form a conjecture and then try to prove it.

(c) For the \(3\ \times\ 3\) matrix A in part (a), use the software to find \(A^2,\ A^3,\ A^4,\ .\ .\ .\) Repeat for the matrices that you constructed in part (b). If you discern a pattern, form a conjecture and then try to prove it.