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Jordan Structures of Totally Nonnegative Matrices

Published online by Cambridge University Press:  20 November 2018

Shaun M. Fallat  and
Michael I. Gekhtman
Shaun M. Fallat
Affiliation:
Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan, S4S 0A2 e-mail: sfallat@math.uregina.ca
Michael I. Gekhtman
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, IN. 46556-5683 U.S.A. e-mail: Michael.Gekhtman.1@nd.edu
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Abstract

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An $n\times n$ matrix is said to be totally nonnegative if every minor of $A$ is nonnegative. In this paper we completely characterize all possible Jordan canonical forms of irreducible totally nonnegative matrices. Our approach is mostly combinatorial and is based on the study of weighted planar diagrams associated with totally nonnegative matrices.

Keywords

15A2115A4805C38totally nonnegativematricesplanar diagramsprincipal rankJordan canonical form
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 57 , Issue 1 , 01 February 2005 , pp. 82 - 98
Copyright
Copyright © Canadian Mathematical Society 2005

References

[1] Ando, T., Totally positive matrices. Linear Algebra Appl. 90 (1987), 165–219.Google Scholar
[2] Berenstein, A., S. Fomin and Zelevinsky, A., Parameterizations of canonical bases and totally positive matrices. Adv. Math. 122 (1996), 49–149.Google Scholar
[3] Brenti, F., Combinatorics and total positivity. J. Combin. Theory Ser. A 71 (1996), 175–218.Google Scholar
[4] Cryer, C. W., Some properties of totally positive matrices. Linear Algebra and Appl. 15 (1976), 1–25.Google Scholar
[5] Fallat, S. M., Bidiagonal factorizations of totally nonnegative matrices. Amer.Math. Monthly 109 (2001), 697–712.Google Scholar
[6] Fallat, S. M., Gekhtman, M. I., and Johnson, C. R., Spectral structures of irreducible totally nonnegative matrices. SIAM J. Matrix Anal. Appl. 22 (2000), 627–645.Google Scholar
[7] Fomin, S. and Zelevinsky, A., Total positivity: tests and parameterizations. Math. Intelligencer 22 (2000), 23–33.Google Scholar
[8] Fomin, S. and Zelevinsky, A., Double Bruhat cells and total positivity. J. Amer. Math. Soc. 12 (1999), 335–380.Google Scholar
[9] Gantmacher, F. R. and Krein, M. G., Sur les matrices complement non-negatives et oscillatories. Comp.Math. 4 (1937), 445–476.Google Scholar
[10] Gantmacher, F. R. and Krein, M. G., Oscillation matrices and kernels and small vibrations of mechanical systems, AMS, Providence, RI, 2002.Google Scholar
[11] Gasca, M. and Micchelli, C. A., eds. Total positivity and its applications, Kluwer Academic, Dordrecht, 1996.Google Scholar
[12] Gasca, M. and Pe˜na, J.M., On factorizations of totally positive matrices. In: Total positivity and its applications, Kluwer Academic, Dordrecht, 1996. pp. 109–130.Google Scholar
[13] Gessel, I. and Viennot, G., Binomial determinants, paths, and hook length formulae. Adv. in Math. 58 (1985), 300–321.Google Scholar
[14] Loewner, C., On totally positive matrices. Math. Z. 63 (1955), 338–340.Google Scholar
[15] Karlin, S., Total positivity, I, Stanford University Press, Stanford, 1968.Google Scholar
[16] Karlin, S. and McGregor, J., Coincidence probabilities. Pacific J. Math. 9 (1959), 1141–1164.Google Scholar
[17] Lusztig, G., Total positivity in reductive groups. In: Lie theory and geometry, Birkhäuser, Boston, MA, 1994, pp. 531–568.Google Scholar
[18] Whitney, A., A Reduction theorem for totally positive matrices. J. Analyse Math. 2 (1952), 88–92.Google Scholar
[19] Shu-fang, Xu, An introduction to inverse algebraic eigenvalue problems, Peking University Press, Beijing, 1998.Google Scholar