Prove or disprove, all eigenvalues of a real symmetric matrix are non-negative.

Question

I tried to find an answer for this question, but what I found was a classification of general matrices (i.e. definite, semi-definite and indefinite). I want to know more specifically about symmetric matrices.

Answer 1

Look at a $2 \times 2$ diagonal matrix with $1$ and $-1$ on the diagonal.

Answer 2

Hint: What is the simplest subset of symmetric matrices you can think of (especially when it comes to determining eigenvalues)? Do any of those have negative eigenvalues?