A basic doubt on some submatrix consisting of transient states of a stochastic matrix

Question

Consider a submatrix of a stochastic matrix where all the states are transient. So, $ Q^n\to 0$ which is clear to me. But why this implies that all of the eigen values of $Q$ have absolute values strictly less than 1.

Answer 1

Let $v$ be an eigenvector for $\lambda$, then $Q^n\to0$ implies $v^T\, Q^n\, v\to0$. But $v^T\, Q^n\, v=v^T \lambda^n v=\lambda^n \|v\|^2,$ and this converges to zero only if $|\lambda|<1$.