A note on an inverse problem for nonnegative matrices

$M$-matrices satisfy Newton’s inequalities

“…The condition (6) follows simply from the fact that tr(A k ) is the kth moment of the eigenvalue sequence of A, while the condition (7) is due to Loewy and London [10] and, independently, Johnson [7].…”

On Open Problems of Nonnegative Inverse Eigenvalues Problem

“…pdf). Readers also may refer to [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] for some previous results. In some articles, some necessary conditions and sufficient conditions for the three problems above have been given under some small dimension or special cases [7].…”

“…A short list of references giving various necessary or sufficient conditions includesBarrett and Johnson (1984), Boyle and Handelman (1991), Friedland (1978), Friedland and Melkman (1979), Loewy and London (1978), de Oliveira (1983), and Reams (1996). The difficulty is that the necessary condition is usually too general and the sufficient condition too specific.…”

Symmetric nonnegative realization of spectra

“…For n ≥ 5 the problem remains unsolved. In the general case, when the possible spectrum Λ is a set of complex numbers, the problem has only been solved for n = 3 by Loewy and London [11]. The complex cases n = 4 and n = 5 have been solved for matrices of trace zero by Reams [17] and Laffey and Meehan [10], respectively.…”