Given a real linear system ($\mathbf{A}\mathbf{x} = \mathbf{b}$), is there any result regarding the positiveness of the solution $\mathbf{x}^*$ considering that $\mathbf{A}$ is diagonally dominant? (see http://en.wikipedia.org/wiki/Diagonally_dominant_matrix for further details about diagonally dominant matrices).
EDIT: Thank you for your quick response and clear answer. Let me add an additional constraint required which is that all elements of $\mathbf{A}$ are positive.
If you make no hypotheses on the positivity/negativity of the entries of $A$ and $b$, nothing can be proved in general: to see this, just pre- or post-multiply by diagonal matrices with $\pm 1$ on the diagonal. This won't affect diagonal dominance, but will change the signs.
The most common case is the one in which you have $A_{ii}>0$ and $A_{ij}<0$ for $i\neq j$. Under this hypothesis + diagonal dominance, your matrix is an M-matrix (as suggested in M. Lin's answer already) and its inverse is positive (hence $x=A^{-1}b$ is positive whenever $b$ is positive).
Strict positivity of the off-diagonal entries can be replaced by a weaker hypothesis (irreducibility), and $\geq$-based versions of these results exist as well.
Many related results are stated in Chapter 9 of Hogben, Handbook of linear algebra. A more comprehensive reference book is Berman-Plemmons, Nonnegative Matrices in the Mathematical Sciences.
If $A$ is an M-matrix, $b$ is a positive vector, then the solution is positive. There is a close relation between M-matrices and diagonally dominant matrices, that is, $A$ is an M-matrix iff there is a positive diagonal matrix $P$ such that $PA$ is diagonally dominant.
