Suppose A is a stochastic matrix and x is an eigenvector with eigenvalue _= 1. Show

Chapter 6, Problem 14

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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.

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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 ,

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