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}, }