finding positive vector that multiplied by matrix give positive vector

Question

Hi, I'm looking to solve the following mathematical problem

I have a given matrix $A$. i want to know if there is a vector $x$ that satisfy two conditions:

is there a mathematical name for that kind of vector? (positive vector?)

I do not know if it has a name, but I have seen it used a couple of times in M-matrix literature. Is your $A$ perchance a Z-matrix? — Federico Poloni, Commented Jan 25, 2012 at 11:19
I'm curios about the general case, but in the case i research now its a Z-matrix / L-matrix (the diagonal is positive and non-diagonal are negative or zero). @Fredrico thanks for the definition. it describes well the A matrix. — iko, Commented Jan 25, 2012 at 11:35
If you have $x$, you can scale it so that all coordinates of $x$ and $Ax$ are greater than $1$. Thus, the existence of $x$ can be expressed in terms of linear programming as feasibility of the system $\vec x\ge\vec1$, $Ax\ge\vec1$. In particular, the linear programming duality may shed some light on the existence of $x$. — Emil Jeřábek, Commented Jan 25, 2012 at 11:52
Did not know before the magical abilities of Linear programming. solved my problems with tools from that field. @Emil - thanks. — user20881, Commented Jan 25, 2012 at 15:22
If your matrix is a Z-matrix, then such an $x$ can also be chosen as the eigenvector relative to the smallest eigenvalue. This might be faster, especially for large matrices. — Federico Poloni, Commented Jan 25, 2012 at 15:29

Federico Poloni · Accepted Answer · 2012-01-25 11:32:01Z

2

For the vector, not that I know. But a matrix $A$ for which such a $x$ exists is called semipositive.

Some useful references as starting points for a literature search: http://www.tandfonline.com/doi/abs/10.1080/03081089408818329, and the book by Berman and Plemmons Nonnegative matrices in the mathematical sciences.

answered Jan 25, 2012 at 11:32

Federico Poloni

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