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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:

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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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up vote 5 down vote accepted

If all entries of $A^{-1}$ are positive numbers, then $A$ has the property you desire.

(Edit: This condition is both sufficient and necessary. If one entry $A^{-1}_{ij}$ is negative, then the choice $b=e_j$ will make the $i$-th component of $x=A^{-1}b$ equal to $A^{-1}_{ij}$, but this is negative.)

The following argument shows that your particular example matrix does have this property: your matrix $A$ has the form

$A = 1 - R F$

where $R=\mathrm{diag}(\rho_1,\rho_2,\dots,\rho_n)$ is the diagonal matrix of reflectivities and $F$ the matrix of form factors. The physical interpretation of $R$ and $F$ makes sure that the norm of $RF$ will be smaller than 1, so that $A^{-1}$ can be expressed as a geometric series

$A^{-1} = (1- RF)^{-1} = 1 + RF + (RF)^2 + (RF)^3 + \dots$

Now, since all entries of $RF$ are positive numbers, this formula implies that the entries of $A^{-1}$ are positive numbers as well.

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