## ON NONNEGATIVE INVERSE EIGENVALUES PROBLEMS

Published: 2019-10-24

Page: 155-175

DANA MAWLOOD MUHAMMED *

Department of Mathematics, Faculty of Science and Art, Gaziantep University, Gaziantep-27310-Turkey.

NECATI OLGUN *

Department of Mathematics, Faculty of Science and Art, Gaziantep University, Gaziantep-27310-Turkey.

MUDHAFAR FATTAH HAMA

Department of Mathematics, University of Slaimani, Sulaimani, Iraq.

*Author to whom correspondence should be addressed.

### 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.

#### 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 https://ikprress.org/index.php/AJOMCOR/article/view/4744

### 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.