Abstract
In this paper, sufficient conditions for the existence and construction of nonnegative matrices with prescribed elementary divisors for a list of complex numbers with nonnegative real part are obtained, and the corresponding nonnegative matrices are constructed. In addition, results of how to perturb complex eigenvalues of a nonnegative matrix while keeping its elementary divisors and its nonnegativity are derived.
Highlights
In this paper, we consider the nonnegative inverse elementary divisors problem (NIEDP)which asks to find necessary and sufficient conditions under which the polynomials (λ − λ1 )n1,(λ − λ2 )n2, . . . , (λ − λk )nk, n1 + n2 + · · · + nk = n, are the elementary divisors of an n × n entrywise nonnegative matrix A [1,2]
Given a list of complex numbers Λ, if there exists a nonnegative matrix A with spectrum Λ, we say that Λ is realizable and that A realizes Λ
The nonnegative inverse eigenvalue problem arises from many areas such as differential equations, functional spaces, mechanics, geophysics, engineering, economy, Markov chains, among others
Summary
We consider the nonnegative inverse elementary divisors problem (NIEDP). The NIEDP contains the nonnegative inverse eigenvalue problem (NIEP), which asks to find necessary and sufficient conditions for a list of complex numbers Λ = {λ1 , . Given a list of complex numbers Λ, if there exists a nonnegative matrix A with spectrum Λ, we say that Λ is realizable and that A realizes Λ This problem has been studied by several authors [3,4,5,6,7,8,9,10,11,12,13,14]. If the matrix is asked to have a prescribed canonical Jordan form, it is called the nonnegative inverse elementary divisors problem.
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