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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

$M$-matrices satisfy Newton's inequalities


Author: Olga Holtz
Journal: Proc. Amer. Math. Soc. 133 (2005), 711-717
MSC (2000): Primary 15A42; Secondary 15A15, 15A45, 15A48, 15A63, 05E05, 05A10, 05A17, 05A19, 26D05, 65F18
Published electronically: August 24, 2004
MathSciNet review: 2113919
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Abstract: Newton's inequalities $c_n^2 \ge c_{n-1}c_{n+1}$ are shown to hold for the normalized coefficients $c_n$ of the characteristic polynomial of any $M$- or inverse $M$-matrix. They are derived by establishing first an auxiliary set of inequalities also valid for both of these classes. They are also used to derive some new necessary conditions on the eigenvalues of nonnegative matrices.


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Additional Information

Olga Holtz
Affiliation: Institut für Mathematik, MA 4-5, Technische Universität Berlin, D-10623 Berlin, Germany
Address at time of publication: Department of Mathematics, University of California-Berkeley, 821 Evans Hall, Berkeley, California 94720
Email: holtz@math.TU-Berlin.DE, holtz@math.berkeley.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07576-8
PII: S 0002-9939(04)07576-8
Keywords: $M$-matrices, Newton's inequalities, immanantal inequalities, generalized matrix functions, quadratic forms, binomial identities, nonnegative inverse eigenvalue problem.
Received by editor(s): March 20, 2003
Received by editor(s) in revised form: November 21, 2003
Published electronically: August 24, 2004
Additional Notes: The author is on leave from the Department of Computer Science, University of Wisconsin, Madison, WI 53706, and is supported by the Alexander von Humboldt Foundation.
Communicated by: Lance W. Small
Article copyright: © Copyright 2004 American Mathematical Society