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## Homework Statement

When trying to solve a question about parameter independence of certain aspects of the Jacobian of a real valued function on a manifold I came to the point where I have to show the following:

Let A be a matrix, J be the Jacobian of an orthogonal transformation (I suppose we can assume non-reflection) then define B to be B= J

^{T}AJ, where A and B are real symmetric. I have to show that both B and A have same number of negative eigenvalues.

## The Attempt at a Solution

This problem I think translates into following:

Both A and B are real symmetric so suppose their diagonal forms are respectively D

_{1}and D

_{2}(ie matrices which have eigenvalues for the diagonal entries). Then there is a unitary transformation U such that D

_{2}= U

^{T}D

_{1}U (an be shown by direct calculation). I have to now show that the number of negative diagonal entries on each D is the same. It seems logical at first because U is an orthogonal matrix but direct calculation does not yield the answer. Moreover when I try to put it into a geometric setting (i.e rotation of vectors) it seems wrong. Am I doing something wrong? edit: I think this might be a wrong approach though because I haven't used the fact that there is a Jacobian as a part of U

Thanks.

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