Are the eigenvectors of positive definite matrix have all positive coordinates?
Oops, sorry. I misread your OP. I wouldn't think the components of an EV of a positive-definite matrix would have to be all positive. In fact, any non-zero scalar multiple of an eigenvector is an eigenvector. Hence, if all the components do happen to be positive, just multiply by a negative scalar, and you've got negative components.
Counterexample
$\displaystyle \begin{bmatrix}{1}&{0}\\{0}&{1}\end{bmatrix}\begin {bmatrix}{-1}\\{-1}\end{bmatrix}=1\begin{bmatrix}{-1}\\{-1}\end{bmatrix}$
Fernando Revilla
Edited: Sorry, I didn' see Ackbeet's post.