(adsbygoogle = window.adsbygoogle || []).push({}); 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 e_{1}...e_{n}and scalars a_{1}...a_{n}such that for i between 1 and n:

Pe_{i}= a_{i}Qe_{i}

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 e_{1}...e_{n}with associated eigenvalues a_{1}...a_{n}s.t

inv(S)*P*inv(S)e_{i}= a_{i}e_{i}for i = 1..n

(possibly dodgy step coming up!)

Now consider the transformation inv(S)*P*inv(S) restricted to span(e_{i})

On this subspace of V, inv(S)*P*inv(S) = a_{i}*I

pre-multiply by S, post multiply by S

to get P = a_{i}*Q on this span

so Pe_{i}= a_{i}Qe_{i}

Is that correct?

Thanks.

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# Question about adjoint transformations- is this a valid proof

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