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Zoe-b
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Homework Statement
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
Homework Equations
Spectral theorem.
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.