Calculate lim n->infinity A^n for large n where
$\displaystyle \left[ \begin{array}{cccc} 1/2 & 0 & 0 \\ 1/2 & 1 & 3/4 \\ 0 & 0 & 1/4\end{array}\right]$
Hi
Another solution has been given in this thread.
The alternate but more mechanical way of doing this is:
Diagonalize the matrix: That is find an invertible P such that $\displaystyle P^{-1}AP = D$. From this we see that $\displaystyle P^{-1}A^n P = D^n$.
Here the eigenvalues are $\displaystyle 1,\frac12, \frac14$
With $\displaystyle P = \begin{pmatrix} 0 & 1 & 0\\ 1 & -1 & 0 \\ 0 & -1 & 3 \end{pmatrix}$
$\displaystyle P^{-1}AP = D = \begin{pmatrix} 1 & 0 & 0\\ 0 & \frac12 & 0 \\ 0 & 0 & \frac14 \end{pmatrix}$
$\displaystyle P^{-1}A^n P = D^n = \begin{pmatrix} 1 & 0 & 0\\ 0 & \left(\frac12\right)^n & 0 \\ 0 & 0 & \left(\frac14\right)^n \end{pmatrix} $
So
$\displaystyle \lim_{n \to \infty}P^{-1}A^n P = \lim_{n \to \infty} \begin{pmatrix} 1 & 0 & 0\\ 0 & \left(\frac12\right)^n & 0 \\ 0 & 0 & \left(\frac14\right)^n \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0\\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} $
$\displaystyle \lim_{n \to \infty}A^n = P \begin{pmatrix} 1 & 0 & 0\\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} P^{-1}$