If $M$ is positive definite, $H$ is self-adjoint. Now consider the minimization problem:$\min_{x\neq 0}\frac{(x,Hx)}{(x,Mx)}.$ Note that this functional is homogeneous of degree 0. So we can just search the minimum on the unite sphere. And because of the continuity of the functional and that the unit sphere of a finite dimensional space is compact. We can find a solution $f$, and by further calculation(variational method), we can show that $f$ satisfies:$Hf=bMf,\qquad \hbox{where $b=\frac{(f,Hf)}{(f,Mf)}$}.$ Now here's my question. Continue to search the minimum in the subspace $\{y:(y,Mf)=0\}$. By the same continuity and compactness argument, the solution exists. Denote it $g$, i.e.$\min_{y\neq 0, \ (y,Mf)=0}\frac{(y,Hy)}{(y,My)}=\frac{(g,Hg)}{(g,Mg)}$ How can I show that $g$ satisfies $Hg=cMg\qquad \hbox{where $c=\frac{(g,Hg)}{(g,Mg)}$}?$
It's probably quite simple... I might think it to be too complicated. Anyone who can give me a hint?
I've just realized that one can always change this kind of generalized Rayleigh Quotient back to the standard Rayleigh Quotient. By changing back and using what has already been established about the relation between minimizing Rayleigh Quotient and the eigenvector of a Hermitian (if in real case, symmetric), I can derive the required equality. But it seems I'm taking a detour. Is there any way to make it more straightforward?