# matrix-theoretic terminology query

Is there an accepted term for the following property?

Let $A$ be a real matrix such that all entries of the eigenvector corresponding to the least eigenvalue have the same sign.

NOTES: (1) The case of multiple eigenvalues might be messy, I don't mind disregarding it at this stage. (2) irreducible Z-matrices, negatives of eventually positive matrices, and inverse-positive matrices all have this property - what I'm looking for is a more general name.

If no standard name comes up, what do you think of the name "minpositive"?

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Since you asked, my uneducated and subjective opinion is that I don't think much of "minpositive". You might consider "firstQ" for eigenvectors in the first quadrant (first orthant actually) and then something like "minfirstQ" for the type of matrix which I find more descriptive. Then again, you might think as much of "minfirstQ" as I do of "minpositive". Gerhard "Matrices By Any Other Name..." Paseman, 2012.04.23 –  Gerhard Paseman Apr 24 '12 at 1:03
so this is in a sense, like a Perron-Frobenius property, but only for the least eigenvalues (so in a way, Perron-Frobenius for the inverse)... –  Suvrit Apr 24 '12 at 4:42
@Suvrit: for the negative ($-A$), rather than the inverse. I think "least" should be understood as "leftmost in the complex plane", otherwise the property does not hold for all Z-matrices as the OP claims. –  Federico Poloni Apr 24 '12 at 7:49
It's supposed to be a common generalization of both cases... –  Felix Goldberg Apr 24 '12 at 8:24
Goldbergian matrices. –  Brendan McKay Apr 24 '12 at 13:27