We can show that, for an n n stochastic matrix, 1 l is an eigenvalue and the remaining eigenvalues must satisfy j 1 j 2, , n Show that if A is an n n stochastic matrix with the property that Ak is a positive matrix for some positive integer k, then j 1 j 2, , n ...

The eigenvalues of A k are k 1 1 k 2 k n Clearly k j 1 for j ...

We can show that, for an n n stochastic matrix, 1 = l is an eigenvalue

Question:

We can show that, for an n × n stochastic matrix, λ1 = l is an eigenvalue and the remaining eigenvalues must satisfy
|λj| ≤ 1 j = 2,..., n
Show that if A is an n × n stochastic matrix with the property that Ak is a positive matrix for some positive integer k, then
|λj| < 1 j = 2,..., n