Spectral theorem for self-adjoint linear transformations

[rainwyz0706]

Let P,Q be self-adjoint linear transformations from V to V, Q is also positive-definite. Deduce that there exist scalars λ1 , . . . , λn and linearly independent vectors e1 , . . . , en in V such that, for i, j = 1, 2, . . . , n:
(i) P ei = λi Qei ;
(ii) <P ei , ej > = δi j λi ;
(iii) <Qei , ej > = δi j .
I could use the spectral theorem to find an orthonormal basis ei for P and Q separately, but how can I connect them together? Could anyone give me some hint? Any help is greatly appreciated!

 

[e(ho0n3]

Let Ei be the linear transformations that project into the subspace spanned by ei. Then (i) is essentially saying that

$$P = \sum \lambda_i QE_i = Q \sum \lambda_i E_i$$.

I hope I'm not giving away too much here.