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MATRICESTOTALLYPOSITIVERELATIVETOATREE ELA

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ELA

MATRICES TOTALLY POSITIVE RELATIVE TO A TREE

∗

CHARLES R. JOHNSON†, ROBERTO S. COSTAS-SANTOS‡, AND BORIS TADCHIEV§

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.

Key words. Totally positive matrices, Sylvester’s identity, Graph theory, Spectral theory.

AMS subject classifications.15A18, 94C15.

∗Received by the editors November 1, 2007. Accepted for publication March 22, 2009. Handling Editor: Bryan L. Shader. This research was conducted during the summer of 2007 as part of the College of William and Mary REU program and was supported by NSF grant DMS-03-53510.

†Department of Mathematics, College of William and Mary, Williamsburg, VA 23187 ([email protected]).

‡Department of Mathematics, College of William and Mary, Williamsburg, VA 23187

([email protected]). The research of RSCS has been supported by Direcci´on General de Inves- tigaci´on del Ministerio de Educaci´on y Ciencia of Spain under grant MTM 2006-13000-C03-02.

§Undergraduate, Binghamton University, Binghamton, NY 13902 ([email protected]).

Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 18, pp. 211-221, April 2009

http://math.technion.ac.il/iic/ela

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