# Maximising a Rayleigh quotient over a subspace

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.

If $z=M^{-1/2}Qx$, then $$K=\max_{z\in S}\frac{z^TMz}{z^Tz},$$ where $S$ is the range of $M^{-1/2}Q$.