Matrices Totally Positive Relative to a Tree, II
Costas-Santos, R. S. and Johnson, C. R. Linear Algebra and its Applications 505 (2016), 1 — 10Abstract
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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@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}, }