matrices satisfy Newton's inequalities
Author:
Olga Holtz
Journal:
Proc. Amer. Math. Soc. 133 (2005), 711717
MSC (2000):
Primary 15A42; Secondary 15A15, 15A45, 15A48, 15A63, 05E05, 05A10, 05A17, 05A19, 26D05, 65F18
Published electronically:
August 24, 2004
MathSciNet review:
2113919
Fulltext PDF Free Access
Abstract 
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Abstract: Newton's inequalities are shown to hold for the normalized coefficients of the characteristic polynomial of any  or inverse 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.
 1.
R.
B. Bapat, Multinomial probabilities, permanents
and a conjecture of Karlin and Rinott, Proc.
Amer. Math. Soc. 102 (1988), no. 3, 467–472. MR 928962
(89k:15008), http://dx.doi.org/10.1090/S00029939198809289629
 2.
Abraham
Berman and Robert
J. Plemmons, Nonnegative matrices in the mathematical
sciences, Academic Press [Harcourt Brace Jovanovich Publishers], New
York, 1979. Computer Science and Applied Mathematics. MR 544666
(82b:15013)
 3.
F.
R. Gantmacher, The theory of matrices. Vol. 1, AMS Chelsea
Publishing, Providence, RI, 1998. Translated from the Russian by K. A.
Hirsch; Reprint of the 1959 translation. MR 1657129
(99f:15001)
 4.
Olga
Holtz and Hans
Schneider, Open problems on GKK 𝜏matrices, Linear
Algebra Appl. 345 (2002), 263–267. MR
1883278, http://dx.doi.org/10.1016/S00243795(01)00492X
 5.
Roger
A. Horn and Charles
R. Johnson, Topics in matrix analysis, Cambridge University
Press, Cambridge, 1994. Corrected reprint of the 1991 original. MR 1288752
(95c:15001)
 6.
Gordon
James, Charles
Johnson, and Stephen
Pierce, Generalized matrix function inequalities on
𝑀matrices, J. London Math. Soc. (2) 57
(1998), no. 3, 562–582. MR 1659833
(2000b:15005), http://dx.doi.org/10.1112/S0024610798005870
 7.
Charles
R. Johnson, Row stochastic matrices similar to doubly stochastic
matrices, Linear and Multilinear Algebra 10 (1981),
no. 2, 113–130. MR 618581
(82g:15016), http://dx.doi.org/10.1080/03081088108817402
 8.
Thomas
J. Laffey, Inverse eigenvalue problems for matrices, Proc.
Roy. Irish Acad. Sect. A 95 (1995), no. suppl.,
81–88. The mathematical heritage of Sir William Rowan Hamilton
(Dublin, 1993). MR 1649820
(99h:15011)
 9.
Thomas
J. Laffey and Eleanor
Meehan, A refinement of an inequality of Johnson, Loewy and London
on nonnegative matrices and some applications, Electron. J. Linear
Algebra 3 (1998), 119–128 (electronic). MR 1637415
(99f:15031)
 10.
Raphael
Loewy and David
London, A note on an inverse problem for nonnegative matrices,
Linear and Multilinear Algebra 6 (1978/79), no. 1,
83–90. MR
0480563 (58 #722)
 11.
Olvi
L. Mangasarian, Nonlinear programming, Classics in Applied
Mathematics, vol. 10, Society for Industrial and Applied Mathematics
(SIAM), Philadelphia, PA, 1994. Corrected reprint of the 1969 original. MR 1297120
(95j:90005)
 12.
Henryk
Minc, Nonnegative matrices, WileyInterscience Series in
Discrete Mathematics and Optimization, John Wiley & Sons Inc., New
York, 1988. A WileyInterscience Publication. MR 932967
(89i:15001)
 13.
Constantin
P. Niculescu, A new look at Newton’s inequalities,
JIPAM. J. Inequal. Pure Appl. Math. 1 (2000), no. 2,
Article 17, 14 pp. (electronic). MR 1786404
(2001h:26020)
 1.
 Bapat, R. B. Multinomial probabilities, permanents and a conjecture of Karlin and Rinott. Proc. Amer. Math. Soc. 102 (1988), no. 3, 467472. MR 0928962 (89k:15008)
 2.
 Berman, Abraham; Plemmons, Robert J. NONNEGATIVE MATRICES IN THE MATHEMATICAL SCIENCES. Computer Science and Applied Mathematics. Academic Press [Harcourt Brace Jovanovich, Publishers], New YorkLondon, 1979. MR 0544666 (82b:15013)
 3.
 Gantmacher, F. R. THE THEORY OF MATRICES. VOL. 1. Translated from the Russian by K. A. Hirsch. Reprint of the 1959 translation. AMS Chelsea Publishing, Providence, RI, 1998. MR 1657129 (99f:15001)
 4.
 Holtz, Olga; Schneider, Hans. Open problems on GKK matrices. Linear Algebra Appl. 345 (2002), 263267. MR 1883278
 5.
 Horn, Roger A.; Johnson, Charles R. TOPICS IN MATRIX ANALYSIS. Corrected reprint of the 1991 original. Cambridge University Press, Cambridge, 1994. MR 1288752 (95c:15001)
 6.
 James, Gordon; Johnson, Charles R.; Pierce, Stephen. Generalized matrix function inequalities on matrices. J. London Math. Soc. (2) 57 (1998), no. 3, 562582. MR 1659833 (2000b:15005)
 7.
 Johnson, Charles R. Row stochastic matrices similar to doubly stochastic matrices. Linear and Multilinear Algebra 10 (1981), no. 2, 113130. MR 0618581 (82g:15016)
 8.
 Laffey, Thomas J. Inverse eigenvalue problems for matrices. Proc. Royal Irish Acad. 95 A (Supplement) (1995), 8188. MR 1649820 (99h:15011)
 9.
 Laffey, Thomas J.; Meehan, Eleanor. A refinement of an inequality of Johnson, Loewy and London on nonnegative matrices and some applications. Electron. J. Linear Algebra 3 (1998), 119128. MR 1637415 (99f:15031)
 10.
 Loewy, Raphael; London, David. A note on an inverse problem for nonnegative matrices. Linear and Multilinear Algebra 6 (1978/79), no. 1, 8390. MR 0480563 (58:722)
 11.
 Mangasarian, Olvi L. NONLINEAR PROGRAMMING. Classics in Applied Mathematics, 10. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994. MR 1297120 (95j:90005)
 12.
 Minc, Henryk. NONNEGATIVE MATRICES. Wiley, New York, 1988. MR 0932967 (89i:15001)
 13.
 Niculescu, Constantin P. A new look at Newton's inequalities. JIPAM. J. Inequal. Pure Appl. Math. 1 (2000), no. 2, Article 17, 14 pp. (electronic). MR 1786404 (2001h:26020)
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Additional Information
Olga Holtz
Affiliation:
Institut für Mathematik, MA 45, Technische Universität Berlin, D10623 Berlin, Germany
Address at time of publication:
Department of Mathematics, University of CaliforniaBerkeley, 821 Evans Hall, Berkeley, California 94720
Email:
holtz@math.TUBerlin.DE, holtz@math.berkeley.edu
DOI:
http://dx.doi.org/10.1090/S0002993904075768
PII:
S 00029939(04)075768
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
