## ON NONNEGATIVE INVERSE EIGENVALUES PROBLEMS

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* = (*λ*_{1}*, **λ*_{2}*, . . . , **λ** _{n}*) 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

*Asian Journal of Mathematics and Computer Research*,

*26*(3), 155–175. Retrieved from https://ikprress.org/index.php/AJOMCOR/article/view/4744