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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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  • $\begingroup$ 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 $\endgroup$
    – Gerhard Paseman
    Commented Apr 24, 2012 at 1:03
  • $\begingroup$ 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)... $\endgroup$
    – Suvrit
    Commented Apr 24, 2012 at 4:42
  • $\begingroup$ @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. $\endgroup$
    – Federico Poloni
    Commented Apr 24, 2012 at 7:49
  • $\begingroup$ It's supposed to be a common generalization of both cases... $\endgroup$
    – Felix Goldberg
    Commented Apr 24, 2012 at 8:24
  • $\begingroup$ Goldbergian matrices. $\endgroup$
    – Brendan McKay
    Commented Apr 24, 2012 at 13:27

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