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I've never seen the term signature of a matrix before this exercise I'm given, and understand it simply means the number of positive eigenvalues.

Anyway:

Let $A_1, A_2$ be real invertible symmetric $2 \times 2$ matrices. I need to determine the possible values for their signatures, and also prove that if they're not congruent, they're simultaneously congruent to diagonal matrices.

Pretty helpless here..

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    $\begingroup$ No, the signature is not the number of positive eigenvalues. It is the numbers of positive, negative and zero eigenvalues (for a real symmetric matrix): different authors might put these in different orders. However, if the matrix is invertible it's determined by the number of positive eigenvalues. $\endgroup$ Jul 7, 2016 at 17:27
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    $\begingroup$ See Sylvester's law of inertia $\endgroup$ Jul 7, 2016 at 17:33

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Hint: one of $A$ and $B$, let's say $A$, is similar to (and WLOG may be considered equal to) either $I$ or $-I$. You can explicitly solve the equations $$(S^T S)_{1,2} = 0,\ (S^T B S)_{1,2} = 0 $$

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