We can show by induction on that if or and .
Hello,
while dealing with nonhomogeneous equations with constant coefficients I ecountered a following problem - I need to calculate powers of a given matrix (all powers up to n-1):
Is there an easy way to calculate it?
I guess the easiest way is just to do the matrix multiplication to find and . You will then see the pattern for (and if you really want to prove it, you can use induction).
In fact, the nonzero elements of all lie on a single diagonal, namely the diagonal k places above the main diagonal. The elements on this diagonal, reading from top left to bottom right, are
That formula works for . Then for
Edit. Beaten to it by girdav!