Let $M$ be an SPD matrix and let $\Pi=QQ^T$ be the orthogonal projection onto the range of  $Q$ (a "tall" matrix with orthonormal columns). I have an expression in the form
$$\tag{1}
K=\max_{v}\frac{v^T(I-\Pi)v}{v^T(I-\Pi)M^{-1}(I-\Pi)v},
$$
which I would like to express as a maximum of something which has $M$ in the numerator and such that the denominator does not depend on $M$.

Actually I know that the solution should be
$$\tag{2}
K=\max_v\frac{v^T(I-\tilde{\Pi})^TM(I-\tilde{\Pi})v}{v^T(I-\Pi)v},
$$
where $\tilde{\Pi}=Q(Q^TM^{-1}Q)^TQ^TM$ (this pie is the $M$-orthogonal projection onto the range of $Q$). I would appreciate any hint on how to get from (1) to (2) (or vice versa).

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I've tried to get from (2) to (1). First, 
$$(I-\tilde{\Pi})^TM(I-\tilde{\Pi})=M^{1/2}(I-\overline{\Pi})M^{1/2},$$ 
where now 
$$\overline{\Pi}=M^{1/2}Q(Q^TM^{-1}Q)^TQ^TM^{1/2}$$
is the orthogonal projection onto the range of $M^{1/2}Q$. So putting this to (2) and taking $w=M^{1/2}v$ gives
$$
K=\max_w\frac{w^T(I-\overline{\Pi})w}{w^TM^{-1/2}(I-\Pi)M^{-1/2}w},
$$
which has already a sort of $M^{-1}$ in the denominator, but still not in the form of (1) and with a different projector in the numerator.



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P.S.: I've asked a [related question][1] already, but the simply stated solution there does not much help me to get to (2) as the subspace over which the maximum is evaluated depends on $M$.


  [1]: http://mathoverflow.net/questions/167362/maximising-a-rayleigh-quotient-over-a-subspace