I was trying to prove that a Markov matrix always has an eigenvalue 1 but I seem to have proved that all eigenvalues of a Markov matrix are equal to one! Could anyone look at my work and point out the mistake?
Let $P$ be a Markov matrix with all rows summing to 1, i.e. $\sum_j P_{ij} = 1 $ for all $i$. Let $\lambda$ be a right eigenvector of P with eigenvalue $\alpha$. Then
$$ \sum_i \lambda_i P_{ij} = \alpha \lambda_j $$
Let's sum both sides over j:
$$ \sum_{i,j} \lambda_i P_{ij} = \sum_j \alpha \lambda_j $$ which gives
$$ \sum_i \lambda_i = \sum_j \alpha \lambda_j $$ which gives $\alpha = 1$! But there should be eigenvalues that are not equal to 1 as well.