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

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This discussion focuses on calculating powers of a 2x2 matrix without using eigenvectors or the diagonalization method. The key technique involves converting the desired power into its binary representation and utilizing exponentiation by squaring. Specifically, for a power p, the method requires approximately 2*log2(p) multiplications, significantly reducing computational effort compared to naive multiplication. The approach emphasizes calculating intermediate powers like M^2 and M^4, then assembling the final result based on the binary representation.

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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.
 
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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.
 

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