# Real inner product spaces and self adjoint linear transformations

#### Kate2010

1. The problem statement, all variables and given/known data

Let V be a real inner product space of dimension n and let Q be a linear
transformation from V to V .
Suppose that Q is non-singular and self-adjoint. Show that Q−1 is self-adjoint.
Suppose, furthermore, that Q is positive-definite (that is, <Qv,v> > 0 for all non-zero v
in V ). Show that the eigenvalues of Q are positive. Deduce that there exists a positive
self-adjoint linear transformation S from V to V such that S2 = Q.
Now let P be a self-adjoint linear transformation from V to V . Show that S−1PS−1 is self-adjoint. Deduce, or prove otherwise, that there exist scalars f1, . . . , fn and linearly independent vectors e1, . . . , en in V such that, for i, j = 1, 2, . . . , n:

*This is the bit I can't do*
(i) Pei = fiQei;
(ii) <Pei, ej> = dij fi;
(iii) <Qei, ej> = dij .

2. Relevant equations

3. The attempt at a solution

So I'm coming back to maths after a few weeks break for Christmas. I have managed most of this question, it is just the last section that I'm unsure how to attempt. I assume I use previous parts of the question, but I'd be really grateful if someone could give me a push in the right direction as to how.

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