Choleski decomposition of a positive matrix

Question

Let us consider a matrix $\boldsymbol{F}$. We consider its Choleski decomposition,

$ \boldsymbol{F} = \boldsymbol{M} \boldsymbol{M}^T $.

We know that $\boldsymbol{F}$ needs to be positive definite. Let us make a further assumption: $\boldsymbol{F}$ is positive, in the sense that none of its entries is negative.

My question is: given this further hypothesis, is $\boldsymbol{M}$ positive too?

Answer 1

The answer is no. In particular, if we take $$ F = \pmatrix{ 1&1&1\\ 1&2&1/2\\ 1&1/2&9/4 } $$ Then we find $$ M = \pmatrix{1&0&0\\1&1&0\\1&-1/2&1} $$ It is true, however, for all $2 \times 2$ matrices.