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= JTAJ, 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 D1 and D2 (ie matrices which have eigenvalues for the diagonal entries). Then there is a unitary transformation U such that D2 = UTD1U (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