Solution Found!
Suppose A is a stochastic matrix and x is an eigenvector with eigenvalue _= 1. Show
Chapter 6, Problem 14(choose chapter or problem)
Suppose A is a stochastic matrix and x is an eigenvector with eigenvalue _= 1. Show directly (i.e., without reference to the proof of Theorem 3.3) that (1, 1, . . . , 1) x = 0.
Questions & Answers
QUESTION:
Suppose A is a stochastic matrix and x is an eigenvector with eigenvalue _= 1. Show directly (i.e., without reference to the proof of Theorem 3.3) that (1, 1, . . . , 1) x = 0.
ANSWER:Step 1 of 3
If x is an eigenvector of A, then , where is a scalar quantity.
Each column vector of a stochastic matrix will be a probability vector. The Sum of elements of a probability vector will be equal to 1.
The eigenvalues , of a stochastic matrix is
When , then x will be a probability vector with sum of the elements equal to 1.
i.e.,
When ,