Is sum of all entries of negative semidefinite matrix is non positive?

Question

I was reading Maximum principle in elliptic pde. In that Auther used following thing directly with out giving the reason

He has one negative semidefinite matrix then he concluded sum of entries of that matrix is non positive. How to prove this fact?

or is there counterexample?

Thanking You Any Help will be appreciated

Answer 1

Let $v$ be the column vector with only $1$'s as entries, and $A$ a negative semidefinite matrix.

What can you say about $v^TAv$?