Page 104 - Textos de Matemática Vol. 39

94 THOMAS LAFFEY AND HELENA SˇMIGOC
Below are listed some necessary conditions on the list of complex numbers σ = (λ1, λ2, . . . , λn) to be the spectrum of a nonnegative matrix.
(1) The Perron eigenvalue max{|λi|; λi ∈ σ} belongs to σ. (2) The list σ is closed under complex conjugation. (3)sk= ni=1λki≥0.
(4) smk ≤ nm−1skm for k,m = 1,2,...
The first condition listed above follows from the Perron-Frobenius theo- rem, which is the basic theorem in the theory of nonnegative matrices. The last condition was proved by Johnson [8] and independently by Loewy and London [17].
The necessary conditions we presented are sufficient only when the list σ has at most three elements. The solution to the NIEP has been found also for lists with four elements [17], while the problem for lists with five or more elements is still open.
One of the most promising attempts at solving the NIEP is to identify the spectra of certain nonnegative structured matrices with known characteristic polynomials. In the paper we first give an overview of the work on companion matrices and then describe some related classes of matrices. Analysis of com- panion matrices gives us a solution to the NIEP when all the eigenvalues except the Perron eigenvalue have nonpositive real parts. We present a method that enables us to find an estimate for the Perron eigenvalue that we need to realize a list of complex numbers with positive real parts. In the last two sections we present examples that illustrate some problems that we face while solving the NIEP.
2. Companion matrices Letf(x)=xn+p1xn−1+...+pn andlet
⎡010...0⎤
⎢0 0 1 ... .⎥
C(f) = ⎢ . . ... ... 0 ⎥
⎢⎣0 0 ... 0 1⎥⎦ −pn −pn−2 . . . −p2 −p1
be the companion matrix of f(x). In order that C(f) be nonnegative, it is necessary that pi ≤ 0 for i = 1,2,...,n.
f(x) = i=1(x − λi) is nonnegative.
Using companion matrices we can solve the NIEP for real lists where all butthedominanteigenvaluearelessthatorequaltozero.Letσ=(λ,λ ,...,λ )
12n bealistofrealnumberssuchthatλi ≤0fori=2,...,n,and ni=1λi ≥0. Sule˘ımanova [22] showed that σ is the spectrum of a nonnegative matrix and Friedlandn [7] showed that the companion matrix of the polynomial