Solution Found!
a. Let A be a stochastic matrix with positive entries, let x Rn, and let y = Ax. Show
Chapter 6, Problem 15(choose chapter or problem)
a. Let A be a stochastic matrix with positive entries, let x Rn, and let y = Ax. Show that |y1| + |y2| + + |yn| |x1| + |x2| + + |xn| and that equality holds if and only if all the (nonzero) entries of x have the same sign. b. Show that if A is a stochastic matrix with positive entries and x is an eigenvector with eigenvalue 1, then all the entries of x have the same sign. c. Prove using part b that if A is a stochastic matrix with positive entries, then there is a unique probability vector in E(1) and hence dim E(1) = 1. d. Prove that if is an eigenvalue of a stochastic matrix with positive entries, then || 1. e. Assume A is a diagonalizable, regular stochastic matrix. Prove Theorem 3.3.
Questions & Answers
QUESTION:
a. Let A be a stochastic matrix with positive entries, let x Rn, and let y = Ax. Show that |y1| + |y2| + + |yn| |x1| + |x2| + + |xn| and that equality holds if and only if all the (nonzero) entries of x have the same sign. b. Show that if A is a stochastic matrix with positive entries and x is an eigenvector with eigenvalue 1, then all the entries of x have the same sign. c. Prove using part b that if A is a stochastic matrix with positive entries, then there is a unique probability vector in E(1) and hence dim E(1) = 1. d. Prove that if is an eigenvalue of a stochastic matrix with positive entries, then || 1. e. Assume A is a diagonalizable, regular stochastic matrix. Prove Theorem 3.3.
ANSWER:Step 1 of 6
a.
It is given that 
And,
Also,
It is known that a square matrix is said to be a stochastic matrix if each of its column vectors is a probability vector.
To prove that 
