I have a 4x4 matrix $M$ and a 4-length vector $V$, and I want to find $M^k\times V$ for very large $k$. Even if I did exponentiation by squaring, there would be way too many steps involved in terms of halving $k$ simply because it is so large. Is there a better way?
Anything faster than matrix exponentiation by squaring?
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1 Answers
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Write $M = PJP^{-1}$ where $J$ is in Jordan normal form. Then $M^k = PJ^kP^{-1}$, and powers of Jordan normal form matrices are much easier to compute.
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0@user51819 No, not really. As I said earlier, squaring is easy, anything else is harder because you need to square first anyway. – 2012-12-09