I wonder whether the following property holds true: For every real symmetric matrix $S$, which is positive in both senses:

$$forall xin{mathbb R}^n,,x^TSxge0,qquadforall 1le i,jle n,,s_{ij}ge0,$$

then $sqrt S$ (the unique square root among positive semi-definite symmetric matrices) is positive in both senses too. In other words, it is entrywise non-negative.

At least, this is true if $n=2$. By continuity of $Smapstosqrt S$, we may assume that $S$ is positive definite. Denoting

$$sqrt S=begin{pmatrix} a & b \ b & c end{pmatrix},$$

we do have $a,c>0$. Because $s_{12}=b(a+c)$ is $ge0$, we infer $bge0$.