# positivity – Square root of doubly positive symmetric matrices

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 $$bge0$$.