Exercise 8

This exercise explains why $\lambda=1$ is an eigenvalue of a stochastic matrix $A$. To conclude that $\lambda=1$ is an eigenvalue, we need to know that $A-I$ is not invertible.

1. What is the product $S(A-I)$?

2. What is the product $S\evec_1$?

3. Consider the equation $(A-I)\xvec = \evec_1$. Explain why this equation cannot be consistent by multiplying by $S$ to obtain $S(A-I)\xvec = S\evec_1$.

4. What can you say about the span of the columns of $A-I$?

5. Explain why we can conclude that $A-I$ is not invertible and that $\lambda=1$ is an eigenvalue of $A$.