If a matrix is non diagonalizable, what other method can I use to calculate the nth power?

Question

First off, I have this matrix A:

1 0 3
1 0 2
0 5 0

I have calculated the eigenvalues, which are (11-sqrt(141))/2 and (11+sqrt(141))/2. From what I understand, if I don't have 3 distinct eigenvalues then the matrix is not diagonalizable in R. Is this matrix diagonalizable in R?

The second part of my question is: if it's not diagonalizable then what other option do I have for calculating A^n? n is really big, something like the order of 10^12. I need to calculate this for a programming problem.

Answer 1

Check the Jordan decomposition (diagonal matrix + nilpotent). But as pointed out in the comments, if you raise something to the power $10^{12}$, don't expect your computer to be able to handle it.

Answer 2

To calculate the eigenvalues:

$$\det(xI-A)=\begin{vmatrix}x-1&0&-3\\ -1&x&-2\\ 0&-5&x\end{vmatrix}=x^2(x-1)-15-10(x-1)=x^3-x^2-10x-5$$

The above cubic has three rather ugly roots, all of them different...and thus the matrix is diagonalizable.