Calculation of a limit, using left eigenvectors, an eigenvector and a positive matrix

Question

I've tried to calculate the limit of 1 over the Spectral Radius of a positive Matrix A times the Matrix A itself, the whole thing to the power of k, but it went wrong somewhere. My attempt is in the picture below, I hope this is alright. Many thanks in advance for your help

my attempt

Answer 1

I just noticed this, and a fair amount of time has gone by. You clearly recognize what you did wrong and you are actually asking how to go about the proof correctly. There are many approaches. The most obvious approach that comes to mind (without assuming that $A$ is disgonalizable) is the following. Try calculating $$ \lim_{u \to \infty} A^u v $$ for any vector $v$. Then write $$ v = a x + b v_\perp $$ where the second term is the part perpendicular to $x$. Then consider the Frobenius Perron theorem and what this says about $r(A)$ versus $|A v_\perp|$ by considering induced matrix norms.