# Proof: Ratio of matrix traces and difference of traces

$\newcommand{\Tr}{\operatorname{Tr}}$

Am looking for a proof that shows that the minimization of $\frac{\Tr X^TAX}{\Tr X^TBX}$ is equivalent to the minimization of $\Tr X^TAX-\lambda \Tr X^TBX$ for some scalar $\lambda$

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Minimizing $f(X)$ with respect to $X$ is equivalent to first minimizing $f(X)$ with respect to $X$ under the constraint $g(X)=c$ and then minimizing the result with respect to $c$. Thus the minimum with respect to $X$ must be the minimum with respect to $X$ under the constraint $g(X)=c$ for some value of $c$, and to find that you can introduce a Lagrange multiplier. Now you just need to replace the denominator by the constant $c$ to arrive at your result.
@qlinck: The constrained function is $g(X)=\operatorname{Tr}X^TBX$. You can tell because the constrained expression is what the Lagrange multiplier $\lambda$ gets multiplied by. To be honest I don't feel like spelling this all out in detail since you haven't shown any effort of your own so far. – joriki Feb 15 '13 at 11:46