Let $M\in\mathbb{R}^{n\times n}$ be symmetric positive definite and consider a matrix $Q\in\mathbb{R}^{n\times m}$ ($m<n$) with orthonormal columns ($Q^TQ=I$). I'm interested in finding an exact expression for $$ K\equiv\max_{x}\frac{x^Tx}{x^TQ^TM^{-1}Qx} $$ in terms of the Rayleigh quotient of $M$ (no inverse). Equivalently, $K$ can be considered as the maximum of $x^Tx/x^TM^{-1}x$ over $x\in\mathcal{R}(Q)$ (the range of $Q$).
Obviously, if $Q=I$, then $$ \max_x\frac{x^Tx}{x^TM^{-1}x}=\max_y\frac{y^TMy}{y^Ty} $$ simply by substituting $y=M^{-1/2}x$. However, that does not work for $K$ above. Of course, what I can do is $$ K=\max_{y}\frac{y^T(Q^TM^{-1}Q)^{-1}y}{y^Ty}, $$ but I can't see that to be very useful.
What I'm actually looking for is something like this: $$ K=\max_{x\in S_1}\min_{y\in S_2(x)}\frac{y^TMy}{x^Tx}, $$ where $S_1$ and $S_2(x)$ are some suitable subspaces of $\mathbb{R}^{n}$ ($S_2$ generally dependent on $x$). However, I'm not sure how to arrive to this max-min characterisation.
Thanks in advance for any suggestion.
P.S.: I have a suspicion that this might be contained in some book dealing thoroughly with variational characterisations of the eigenvalues of SPD matrices. However, I haven't found any with such a result in my library.
