Question

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.

Answer 1

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.