Prove that all diagonal entries of a negative definite matrix are negative

Question

We know that all principal minors of order one are nonpositive, but is there a way to prove that all of them are negative? I looked at a similar question here but the solution is too vague, so if possible, please explain it in more detail.

Answer 1

If you choose the vector $x=(0,0,\dots,1,\dots)$ with the $1$ in the $k$-th position, then $$ x^TAx=a_{kk} $$ Since $x\neq 0$ you should have $a_{kk}<0$.