Signatures of symmetric $2 \times 2$ matrices.

Question

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..

Answer 1

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 $$(S^T S)_{1,2} = 0,\ (S^T B S)_{1,2} = 0 $$