A = [ [ 4 , -2, 3] , [$\frac{1}{2}$ , 0 , $\frac{1}{2}$] , [-4,$\frac{5}{2}$,-3] ]
Suppose, we know the eigenvalues $\frac{1}{2}$,$-\frac{1}{2}$ and 1 of A and a matrix T with $T^{-1}AT = D$, where D is a diagonal matrix with the eigenvalues as its entries. $T^{-1}$ was also calculated in the exercise.
How can we calculate $lim_{n->\infty}A^n$ from this ?
I tried to use that $A^n$ has eigenvalue $\lambda^n$, if A has eigenvalue $\lambda$, but could not make progress.
Since we have $$A^n=T D^nT^{-1},$$ and $\lim\limits_{n\rightarrow\infty}D^n=\left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1\end{array}\right)$, the limiting expression for $A^n$ will be $$\lim_{n\rightarrow\infty}A^n=T \left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1\end{array}\right)T^{-1}.$$ Even more specifically, $$\left(\lim_{n\rightarrow\infty}A^n\right)_{jk}=T_{j3}(T^{-1})_{3k}.$$