Consider the $n\times n$ matrix $F$ defined by the following expression $$ F=A-\varepsilon B $$ where $A$ is a constant matrix such that $a_{ij}=a>0$ for all $i,j$ and where $B$ is a symmetric matrix such that $b_{ij}\geq 0$ for all $i,j$.

I would like to find sufficient conditions on $A$ and $B$ such that $F$ is positive semi-definite when the scalar $\varepsilon>0$ is small enough.

Of course, this works if $B$ is negative semi-definite. It also works for $n=2$ if the diagonal elements of $B$ are 0. In this case $$ F=\left[\begin{array}{cc} a & a-\varepsilon b\\ a-\varepsilon b & a \end{array}\right] $$is diagonally dominant for $\varepsilon$ small enough. Sadly I'm unable to generalize this zero-diagonal case to $n>2$.