Some results on the symmetric doubly stochastic inverse eigenvalue problem

Document Type : Research Paper

Authors

Department of Mathematics‎, ‎Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice‎, ‎East China Normal University‎, ‎Shanghai‎, ‎200241‎, ‎P‎. ‎R‎. ‎China.

Abstract

‎The symmetric doubly stochastic inverse eigenvalue problem (hereafter SDIEP) is to determine the necessary and sufficient conditions for an $n$-tuple $\sigma=(1,\lambda_{2},\lambda_{3},\ldots,\lambda_{n})\in \mathbb{R}^{n}$ with $|\lambda_{i}|\leq 1,~i=1,2,\ldots,n$‎, ‎to be the spectrum of an $n\times n$ symmetric doubly stochastic matrix $A$‎.
‎If there exists an $n\times n$ symmetric doubly stochastic matrix $A$ with $\sigma$ as its spectrum‎, ‎then the list $\sigma$ is s.d.s‎. ‎realizable‎, ‎or such that $A$ s.d.s‎. ‎realizes $\sigma$‎. ‎In this paper‎, ‎we propose a new sufficient condition for the existence of the symmetric doubly stochastic matrices with prescribed spectrum‎. ‎Finally‎, ‎some results about how to construct new s.d.s‎. ‎realizable lists from the known lists are presented.

Keywords

Main Subjects

References

J. Ccapa and R. L. Soto, On spectra perturbation and elementary divisors of positive matrices, Electron. J. Linear Algebra 18 (2009) 462--481.
M.T. Chu and G.H. Golub, Inverse Eigenvalue Problems: Theory, Algorithms, and Application, Oxford Science Publications, Oxford Univ. Press, 2005.
P.D. Egleston, T.D. Lenker and S.K. Narayan, The nonnegative inverse eigenvalue problem, Linear Algabra Appl. 379 (2004) 475--490.
L. Elsner, R. Nabben and M. Neumann, Orthogonal bases that lead to symmetric non-negative matrices, Linear Algebra Appl. 271 (1998) 323--343.
S.D. Eubanks and J.J. McDonald, On a generalization of Soules bases, SIAM J. Matrix Anal. Appl. 31 (2009), no. 3, 1227--1234.
M. Fang, A note on the inverse eigenvalue problem for symmetric doubly stochastic matrices, Linear Algebra Appl. 432 (2010), no. 11, 2925--2927.
S.K. Hwang and S.S. Pyo, The inverse eigenvalue problem for symmetric doubly stochastic matrices, Linear Algebra Appl. 379 (2004) 77--83.
C. Knudsen and J.J. McDonald, A note on the convexity of the realizable set of eigenvalues for nonnegative symmetric matrices, Electron. J. Linear Algebra 8 (2001) 110--114.
Y.J. Lei, W.R. Xu, Y. Lu, Y.R. Niu and X.M. Gu, On the symmetric doubly stochastic inverse eigenvalue problem, Linear Algebra Appl. 445 (2014) 181--205.
L.F. Martignon, Doubly stochastic matrices with prescribed positive spectrum, Linear Algebra Appl. 61 (1984) 11--13.
J.J. McDonald and M. Neumann, The Soules approach to the inverse eigenvalue problem for nonnegative symmetric matrices of order n5, in: Algebra and Its Applications,
(Proc. International Conference, Algebra and Its Apllications, Ohio Univ. 1999), pp. 387--407, Contemp. Math. 259, Amer. Math. Soc., Providence, RI, 2000.
H. Minc, Nonnegative Matrices, John Wiley & Sons, New York, 1988.
B. Mourad, An inverse problem for symmetric doubly stochastic matrices, Inverse Problems 19 (2003), no. 4, 821--831.
B. Mourad, A note on the boundary of the set where the decreasingly ordered spectra of symmetric doubly stochastic matrices lie, Linear Algebra Appl. 416 (2006) 546--558.
B. Mourad, On a spectral property of doubly stochastic matrices and its application to their inverse eigenvalue problem, Linear Algebra Appl. 436 (2012) 3400--3412.
B. Mourad, H. Abbas and M.S. Moslehian, A note on the inverse spectral problem for symmetric doubly stochastic matrices, Linear Multilinear Algebra 63 (2015) 2537--2545.
 B. Mourad, H. Abbas, A. Mourad, A. Ghaddar and I. Kaddoura, An algorithm for constructing doubly stochastic matrices for the inverse eigenvalue problem, Linear Algebra Appl. 439 (2013), no. 5, 1382--1400.
H. Perfect and L. Mirsky, Spectral properties of doubly-stochastic matrices, Monatsh. Math. 69 (1965) 35--57.
R. Reams, Constructions of trace zero symmetric stochastic matrices for the inverse eigenvalue problem, Electron. J. Linear Algebra 9 (2002) 270--275.
O. Rojo and H. Rojo, Constructing symmetric nonnegative matrices via the fast Fourier transform, Comput. Math. Appl. 45 (2003), no. 10, 1655--1672.
O. Rojo and H. Rojo, Some results on symmetric circulant matrices and on symmetric centrosymmetric matrices, Linear Algebra Appl. 392 (2004) 211--233.
R.L. Soto and O. Rojo, Application of a Brauer theorem in the nonnegative inverse eigenvalue problem, Linear Algebra Appl. 416 (2006), no. 2-3, 844--856.
G.W. Soules, Constructing symmetric nonnegative matrices, Linear Multilinear Algebra 13 (1983), no. 3, 241--251.
W.R. Xu, Y.J. Lei, X.M. Gu, Y. Lu and Y.R. Niu, Comment on:A note on the inverse eigenvalue problem for symmetric doubly stochastic matrices", Linear Algebra Appl. 439 (2013), no. 8, 2256--2262.
X. Zhan, Matrix Theory, Amer. Math. Soc. Providence, RI, 2013.
S. Zhu, T. Gu and X. Liu, Solving inverse eigenvalue problems via Householder and rank-one matrices, Linear Algebra Appl. 430 (2009), no. 1, 318--334.
Bulletin of the Iranian Mathematical Society
Volume 43, Issue 3
June 2017
Pages 853-865

Files

History

  • Receive Date: 25 July 2015
  • Revise Date: 03 March 2016
  • Accept Date: 14 March 2016

Share

How to cite

Statistics