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Am looking for a proof of non-convexity of the quotient of two matrix trace functions as given by $\frac{\operatorname{Tr}X^TAX}{\operatorname{Tr}X^TBX}$, when $TrX^TBX>0$ for two different positive semi-definite matrices , $A$ and $B$ such that $A \neq \alpha B$ for any scalar $\alpha$.

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    Also, it's not clear to me if you are asking about the convexity of the function, or of the set of numbers.2012-04-19

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I'm not exactly sure what you are looking for, but a specific counterexample demonstrates 'non-convexity':

Choose

$ A = \left( \begin{array}{cc} 1 & 0 \\ 0 & 10 \end{array} \right), B = \left( \begin{array}{cc} 10 & 0 \\ 0 & 1 \end{array} \right), X(t) = \left( \begin{array}{cc} 1 & 0 \\ 0 & t \end{array} \right). $ Some manipulation yields: $\phi(t) = \frac{\mathbb{tr}(X^T A X)}{\mathbb{tr}(X^T B X)} = \frac{10 t^2+1}{t^2+10}.$ The function $\phi$ is not convex (for example, $\phi(5) > \frac{1}{2}(\phi(0)+\phi(10))$.