1. The problem statement, all variables and given/known data Q is an invertible self-adjoint linear transformation on an inner product space V. Suppose Q is positive definite. I have already shown that inv(Q) is self-adjoint, that all eigenvalues of Q are positive, so there exists S s.t. S^2 = Q. Now suppose P is any self-adjoint linear transformation on V. I have shown inv(S)*P*inv(S) is also self adjoint. The bit I'm trying to do is: deduce or prove otherwise that there are linearly independant vectors e1...en and scalars a1...an such that for i between 1 and n: Pei = aiQei 2. Relevant equations Spectral theorem. 3. The attempt at a solution I have been staring at this for ages- I've done this but I'm not sure at all that its valid so if its complete rubbish please let me know! inv(S)*P*inv(S) is self-adjoint so by the spectral theorem there exists an orthonormal (and therefore linearly independent) basis of eigenvectors e1...en with associated eigenvalues a1...an s.t inv(S)*P*inv(S)ei = aiei for i = 1..n (possibly dodgy step coming up!) Now consider the transformation inv(S)*P*inv(S) restricted to span(ei) On this subspace of V, inv(S)*P*inv(S) = ai*I pre-multiply by S, post multiply by S to get P = ai*Q on this span so Pei = aiQei Is that correct? Thanks.