Let Q be a n by n positive definite or positive semi definite matrix and g be a vector in $R^{n}$.
Is there a closed form to get x?
$g^{T}Q^{k}g = x(g^{T}Qg)$
where k is a some integer number.
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$\begingroup$ Please use LaTex $\endgroup$– Riccardo.AlestraDec 5, 2013 at 14:16
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$\begingroup$ If it's positive definite, can't you just use eigendecomposition of $Q^k$ and then divide by $g^TQg$? $\endgroup$– hejsebDec 5, 2013 at 14:33
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$\begingroup$ If the matrix Q is too huge to get eigendecomposition, do you have other suggestions? $\endgroup$– Jaehwan JeongDec 5, 2013 at 14:35
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$\begingroup$ If $Q$ has unique largest eigenvalue $\lambda$, then for almost all $g$, for large $k$ we know that $x$ grows (or decays) like $\lambda^{k}$. $\endgroup$– GEdgarDec 5, 2013 at 14:36
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1$\begingroup$ Isn't $x=\frac{g^TQ^kg}{g^TQg}$ closed enough (provided $g^TQg\neq 0$)? $\endgroup$– Algebraic PavelDec 5, 2013 at 15:12
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