Matrices Totally Positive Relative to a Tree, II

Abstract

If T is a labelled tree, A is totally positive relative to T , principal submatrices of A associated with deletion of pendent vertices of T are P-matrices, and A has positive determinant, then the smallest absolute eigenvalue of A is positive with multiplicity 1 and its eigenvector is signed according to T. This conclusion has been incorrectly conjectured under weaker hypotheses.

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BibTeX

@article {MR3506481,
AUTHOR={Costas-Santos, R. S. and Johnson, C. R.},
TITLE={Matrices totally positive relative to a tree, {II}},
JOURNAL={Linear Algebra Appl.},
FJOURNAL={Linear Algebra and its Applications},
VOLUME={505},
YEAR={2016},
PAGES={1--10},
ISSN={0024-3795},
MRCLASS={15B48 (05C05 05C50 15A18 94C15)},
MRNUMBER={3506481},
ZBL={1337.15025},
MRREVIEWER={Carlos Marijuan},
DOI={10.1016/j.laa.2016.04.021},
URL={http://dx.doi.org/10.1016/j.laa.2016.04.021},
}