ON NONNEGATIVE INVERSE EIGENVALUES PROBLEMS

Main Article Content

DANA MAWLOOD MUHAMMED
NECATI OLGUN
MUDHAFAR FATTAH HAMA

Abstract

Inverse eigenvalue problems constitute an important subclass of inverse problems that arise in the context of mathematical modelling and parameter identification.

The inverse eigenvalue problem for nonnegative matrices has a very simple formulation: given a list L = (λ1, λ2, . . . , λn) of complex numbers, find necessary and sufficient conditions for the existence of an n-square nonnegative matrix A with  spectrum L. This problem is a very difficult one and it remains unsolved for any positive integer n.

In this work, we will reconstruct the nonnegative matrices induced by; Lowey and London for n=3, Reams for n=4 ,5 , Laffey and Meehan for n=5 ; by using Newton’s identities defined in linear algebra by Dan Kalman.  Also, we use Newton’s identities to construct the non negative matrices for n=6,7,8

Keywords:
Eigenvalue problem, inverse eigenvalue problem, Newton’s identities, nonnegative matrix, spectrum.

Article Details

How to Cite
MUHAMMED, D. M., OLGUN, N., & FATTAH HAMA, M. (2019). ON NONNEGATIVE INVERSE EIGENVALUES PROBLEMS. Asian Journal of Mathematics and Computer Research, 26(3), 155-175. Retrieved from http://www.ikprress.org/index.php/AJOMCOR/article/view/4744
Section
Original Research Article

References

Kolmogorov AN. Markov chains with countably many possible states. Bull. Univ. Moscow (A) 3. 1937;1–16. (in Russian)

Suleimanova KR. Stochastic matrices with real eigenvalues. Soviet Math. Dokl. 1949;66:343–345. (in Russian)

Minc H. Nonnegative matrices. John Wiley and Sons, New York; 1988.

Dmitriev N, Dynkin E. On the characteristic numbers of a stochastic matrix. Dokl. Nauk SSSR. 1945;49:159-162.

Dmitriev N, Dynkin E. Eigenvalues of a stochastic matrix. Izv. Akad. Nauk SSSR Ser. Mat. 1946;10: 167-184.

Karpelevich F. On the eigenvalues of a matrix with nonnegative elements. Izv. Akad. Nauk SSSR Ser. Mat. 1951;15:361-383.

Perfect H. On positive stochastic matrices with real characteristic roots. Proc. Cambridge Phil. Soc. 1952;48:271-276.

Ciarlet PG. Some results in the theory of nonnegative matrices. Linear Algebra. Appl. 1968;1:139-152.

Kellogg RB. Matrices similar to a positive or essentially positive matrix. Linear Algebra Appl. 1971;4:191-264.

Perfect H. Methods of constructing certain stochastic matrices. Duke Math. J. 1953;20:395-404.

Salzmann F. A note on eigenvalues of nonnegative matrices. Linear Algebra Appl. 1972;5:329-338.

Fiedler M. Eigenvalues of nonnegative symmetric matrices. Linear Algebra Appl. 1974;9:119-142.

Borobia A. On the nonnegative eigenvalue problem. Linear Algebra Appl. 1995;223-224:131-140.

Loewy R, London D. A note on an inverse problem for nonnegative matrices. Linear and Multilinear Algebra. 1978;6:83-90.

Reams R. An inequality for nonnegative matrices and the inverse eigenvalue problem. Linear and Multilinear Algebra. 1996;41:367-375.

Laffey T, Meehan E. A characterization of trace zero nonnegative 5 × 5 matrices. Linear Algebra Appl. 1999;302-303:295–302.

Xu S. On inverse spectrum problems for nonnegative matrices. Linear and Multilinear Algebra. 1993; 34:353-364.

Radwan N. An inverse eigenvalue problem for symmetric and normal matrices. Linear Algebra Appl. 1996;248:101-109.

Guo Wuwen. Eigenvalues of nonnegative matrices. Linear Algebra Appl. 1997;266:261-270.

Kalman D. A Matrix Proof of Newton’s Identities. Mathematics Magazine. 2000;73:313-315.

Cronin AG. Constructive methods for spectra with three nonzero elements in the nonnegative inverse eigenvalue problem. Linear and Multilinear Algebra. 2018;66(3):435-446.

Cronin AG, Laffey TJ. The diagonalizable nonnegative inverse eigenvalue problem. Special Matrices. 2018;6(1):273-281.

Johnston N, Patterson E. The inverse eigenvalue problem for entanglement witnesses. Linear Algebra and its Applications. 2018;550:1-27.

Meyer CD. Matrix analysis and applied linear algebra. SIAM Press; 2000.

Macdonald IG. Symmetric functions and hall polynomials. 2nd Edition, Oxford Mathematical Monographs, Oxford University Press, USA; 1995.

Egleston PD, Lenker TD, Narayan SK. The nonnegative inverse eigenvalue problem. Linear Algebra Appl. 2004;379:475-490.

Laffey T, Meehan E. A refinement of an inequality of Johnson, Loewy, and London on nonnegative matrices and some applications. Electron. J. Linear Algebra. 1998;3:119-128.

Boyle M, Handelman D. The spectra of nonnegative matrices via symbolic dynamics. Annals of Mathematics. 1991;133(2):249-316.