Matrices Totally Positive Relative to a Tree

Abstract

It is known that for a totally positive (TP) matrix, the eigenvalues are positive and distinct and the eigenvector associated with the smallest eigenvalue is totally nonzero and has an alternating sign pattern. Here, a certain weakening of the TP hypothesis is shown to yield a similar conclusion.

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BibTeX

@article {MR2505131,
AUTHOR={Johnson, C. R. and Costas-Santos, R. S. and Tadchiev, B.},
TITLE={Matrices totally positive relative to a tree},
JOURNAL={Electron. J. Linear Algebra},
FJOURNAL={Electronic Journal of Linear Algebra},
VOLUME={18},
YEAR={2009},
PAGES={211--221},
MRCLASS={15B48 (05C05 05C50 15A18 94C15)},
MRNUMBER={2505131},
ZBL={1171.15021},
MRREVIEWER={Carlos Mariju\'{a}n},
DOI={10.13001/1081-3810.1306},
URL={https://doi.org/10.13001/1081-3810.1306},
}