Let's say I have $3\times3$ matrix $A$, which is known. I'm given $A^x$ and $A^y$. The goal is to determine $A^{xy}$. $x$ and $y$ are unknown. Is it possible?
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1$\begingroup$ If the matrix is diagonalizable powers are easy. Do you know if you can diagonalize your matrix? $\endgroup$– WintermuteDec 18, 2015 at 15:25
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$\begingroup$ @Wintermute Does that mean if the matrix can be Gauss-Jordaned so that everything outside the main diagonal is 0? $\endgroup$– CaptainCodemanDec 18, 2015 at 15:27
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$\begingroup$ @CaptainCodeman no, this is matrix similarity, not row-reduction ("Gauss-Jordaning"). $\endgroup$– Ben GrossmannDec 18, 2015 at 15:30
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1$\begingroup$ Euh...A brute-force solution: since the characteristic polynomial of A is of degree $3,$ we can always compute explicitly (at least theoretically) the powers $A^n,$ so we can conversely obtain the values of $x$ and $y$... Of course this is not what you want. :P $\endgroup$– awllowerDec 18, 2015 at 15:30
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$\begingroup$ @CaptainCodeman I'm not quite sure what you mean by Gauss-Jordaned. A matrix is diagonalizable if you can write it $A=P^{-1}DP$ for some diagonal matrix D. Then we have the nice result $A^x=P^{-1}D^xP$, which is great because it's easy to raise a diagonal matrix to a power. $\endgroup$– WintermuteDec 18, 2015 at 15:32
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