`Ax = b`. I need a way to analyze a square matrix `A` to see if its solution vector `x` will always be positive when `b` is positive.


This question arises from solving the radiosity equation:


<http://www.siggraph.org/education/materials/HyperGraph/radiosity/images/slide11.jpg> (noob alert)


I'm interested to know when A is *incorrect*, which would be when `x` has negative values even though `b` is positive.



How can I characterize the type of solution vector that comes out of a matrix?
If and only if all the entries of $A^{-1}$ are non-negative. 


**Proof:** If $(A^{-1})\_{ij}$ is negative, and $b$ is $1$ in the $j$-th coordinate and very small in every other, then $A^{-1} b$ is negative in the $i$-th component. 


On the other hand, if every entry of $A^{-1}$ is non-negative, then clearly $b$ positive implies $A^{-1} b$ positive.


