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.



Maximising a Rayleigh quotient over a subspace
If $z=M^{-1/2}Qx$, then 
$$K=\max\_{z\in S}\frac{z^TMz}{z^Tz},$$
where $S$ is the range of $M^{-1/2}Q$. 


