Positive solutions only for linear equations

Factual Questions
1

I’ve searched but can’t find out if this problem has been addressed. Does anyone have any references (or thoughts) on the following:

Consider a set of simultaneous linear equations:

a[sub]11[/sub]x[sub]1[/sub] + … + a[sub]1n[/sub]x[sub]n[/sub] = b[sub]1[/sub]
…
a[sub]n1[/sub]x[sub]1[/sub] + … + a[sub]nn[/sub]x[sub]n[/sub] = b[sub]n[/sub]

All a[sub]ij[/sub] >= 0, b[sub]i[/sub] >= 0.

Under what conditions are all the solutions x[sub]i[/sub] > 0. (I’d accept >= here if easier.)

I know there is a result about the number of positive solutions to a polynomial and I’ve thought about the linear programming dual, but neither seems to help.

2

Never mind, I found this result in economics of all places. The conditions are known as the Hawkins-Simon conditions – now I just have to figure out what they mean exactly.

3

The general area that you’re looking for is known as “sign solvable linear systems”. There’s a book, but I don’t know how accessible it is to a non-specialist.