So we have a recurrence relation v$_n$=T v$_{n-1}$ which we can write as v$_n$=T$^n$ v$_0$
We are interested in showing that the ratio of the sum of entries in v$_n$ and v$_{n-1}$ tends to the spectral radius of the matrix T
Written more formally
Let ||$\cdot$|| be the taxi cab norm, that is if v=(v$_{1}$,v$_{2}$, ... ,v$_{k}$) then ||v||=$\sum_{i=1}^k |v$$_i$$|$
Then $\lim\limits_{n \to \infty} \frac{||v_n||}{||v_{n-1}||}$ = Spectral radius of T
I have no idea how to prove this and in the document im reading it in it says
"It is a basic fact of linear homogeneous recurrence relations that a closed-form solution to the recurrence relation can be written down in terms of the eigenvalues of the transition matrix. As a corollary of this, the limiting ratio of terms in the sequence is equal to the spectral radius of the transition matrix."