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!