Sum of positive semidefinite and negative definite matrices

Question

Given symmetric matrices $A$ and $B$, where $A$ is singular and positive semidefinite and $B$ is diagonal and negative definite, is it possible to get conditions such that $A+B$ is positive semidefinite?

Answer 1

Since $A$ is singular, there is a vector $v$ with $Av=0$. Since $B$ is negative definite, we have $\def\<{\langle}\def\>{\rangle}\<v,Bv\><0$, which gives

$$\<v,(A+B)v\>=\underbrace{\<v,Av\>}_{=\,0}+\underbrace{\<v,Bv\>}_{<\,0}<0.$$

Hence $A+B$ cannot be positive semi-definite.