Q:

Elements of a Negative Semidefinite Matrix

Use the definition of a negative definite matrix to show that if A is negative semi-definite:
$$A_{ii} ≤ 0 \ \forall i $$
I know the definition (in terms of quadratic form) and the equivalent rules (principal minors, eigenvalues, etc) but I can't seem to get this proof out! I'm not even quite sure why the question suggests using the definition of a negative definite matrix, other than the fact negative definite implies negative semi-definite. 
Any pointers in the right direction?

A:

Negative semi-definite is equivalent to $x^TAx\le 0$ for all $x$. Now, choose the right $x$ to obtain the desired result.

