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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.

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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.

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  • $\begingroup$ You are fast David, I was just in the middle of posting. $\endgroup$
    – Tony Huynh
    Commented Aug 11, 2010 at 18:59
  • $\begingroup$ A wide class of matrices having nonnegative inverses is that of M-matrices. en.wikipedia.org/wiki/M-matrix . The matrix in the link you posted has the right sign-pattern (i.e., it is a Z-matrix), so you just need to check that the spectral radius of the matrix $\rho_i F_{ij}$ is less than 1 to make sure that your matrix is an M-matrix. $\endgroup$
    – Federico Poloni
    Commented Aug 12, 2010 at 12:22
  • $\begingroup$ @David, thank you. @Federico: Thank you for that insightful comment. $\endgroup$
    – bobobobo
    Commented Aug 12, 2010 at 16:23

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