How to calculate powers of a 2x2 matrix WITHOUT eigenvectors ?

[sid9221]

How do I determine powers of matrices(2x2) without calculating their eigenvectors and doing the pdp^-1 thing ?

Obviously multiplying over and over is not a solution.

 

[lpetrich]

I'll let someone else try doing that without straight multiplication, but even with straight multiplication, there is a way.

If you want only one power, find its binary representation: b0+b1*2+b2*2^2+...

Then calculate powers of the matrix: M^4 = (M^2)^ 2, M^8 = (M^4)^2, etc.
Then assemble the final result: identity * (multiply by M if b0 is 1) * (multiply by M^2 if b1 is 1) * (multiply by M^4 if b2 is 1) * ...

For power p, instead of p multiplications, one has to do around 2*log(2,p) ones.