Our professor showed us the trick, that if a matrix has positive and negative elements on the diagonal, its automatically indefinite.
After analyzing the script, I think the trick must come from the quadratic form:
$Q_A (a) = x'Ax$
And the quadratic form is called indefinite, if there are vectors for x with $Q_A (a) < 0$ and $Q_A (a) > 0$. So it's always possible to plug in a specific vector, e.g. $x = e_i = [1, 0, 0, 0, ...]$ and get any diagonal element.
But then the professors trick only works for symmetric matrices.
My question am I on the right track? Is this where the trick is coming from? Or is it also possible to apply the trick to any matrix (not necessarily symmetric)?
Thanks!