Maxima of $Tr(X^TAX)/Tr(X^TBX)$ if $A,B$ are p.s.d?

Question

What is the maxima of $g(X) = Tr(X^TAX)/Tr(X^TBX)$ if $A,B$ are square, symmetric and positive semi-definite? All matrices are real-valued and X is a rectangular matrix. Also there is a constraint that $Tr(X^TC)=\alpha$ where $\alpha$ is a non-zero scalar. Is it trivial based on the eigen-vectors of $A$ and $B$, or is there more to it?

Answer 1

For positive definite $B$ we have $$ \max_{X\ne 0} \frac{tr(X^T A X)}{tr(X^T B X)} = \max_{tr(Y^T Y) = 1} tr(Y^T B^{-1/2} A B^{-1/2} Y) = \lambda_{max}(B^{-1/2} A B^{-1/2}). $$