So it gives me this matrix
And asks me to compute A^k, where k is any positive integer.
I know I have to use to eigenvectors which I found to be this matrix:
and the diagonal matrix made out of the eigenvalues:
Now I know A^k=P*D^k*P^-1 but when I do this I just get something really messy,
Any help will be appreciated.
Another way: using the euclidean division, . Using the Cayley-Hamilton theorem, . Substituting by in we get the system
Solving and substituting in we get . This is not necessarily a faster method but has the advantage that is also valid for non diagonalizable matrices (differentiating [1] conveniently).