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.



Positive solutions of linear systems with a diagonally dominant matrix
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](https://en.wikipedia.org/wiki/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*. 


