About the persymmetric nonnegative inverse eigenvalue problem in low dimension

The nonnegative inverse eigenvalue problem from the coefficients of the characteristic polynomial. EBL digraphs

The longstanding nonnegative inverse eigenvalue problem (NIEP) is to determine which multisets of complex numbers occur as the spectrum of an entry-wise nonnegative matrix. Although there are some well-known necessary conditions, a solution to the NIEP is far from known. An invertible matrix is called a Perron similarity if it diagonalizes an irreducible, nonnegative matrix. Johnson and Paparella [Linear Algebra Appl. 493 (2016), pp. 281–300] developed the theory of real Perron similarities. Here, we fully develop the theory of complex Perron similarities. Each Perron similarity gives a nontrivial polyhedral cone and convex polytope of realizable spectra (thought of as vectors in complex Euclidean space). The extremals of these convex sets are finite in number, and their determination for each Perron similarity would solve the diagonalizable NIEP, a major portion of the entire problem. By considering Perron similarities of certain realizing matrices of Type I Karpelevič arcs, large portions of realizable spectra are generated for a given positive integer. This is demonstrated by producing a nearly complete geometrical representation of the spectra of four-by-four stochastic matrices. Similar to the Karpelevič region, it is shown that the subset of complex Euclidean space comprising the spectra of stochastic matrices is compact and star-shaped. Extremal elements of the set are defined and shown to be on the boundary. It is shown that the polyhedral cone and convex polytope of the discrete Fourier transform (DFT) matrix corresponds to the conical hull and convex hull of its rows, respectively. Similar results are established for multifold Kronecker products of DFT matrices and multifold Kronecker products of DFT matrices and Walsh matrices. These polytopes are of great significance with respect to the NIEP because they are extremal in the region comprising the spectra of stochastic matrices. Implications for further inquiry are also given.

The nonnegative inverse eigenvalue problem is the problem of finding necessary and sufficient conditions for the existence of an entrywise nonnegative matrix A with prescribed spectrum. This problem remains open for . If the matrix A is required to be normal, the problem will be called the normal nonnegative inverse eigenvalue problem (NNIEP). Sufficient conditions for a list of complex numbers to be the spectrum of a normal nonnegative matrix were obtained by Xu [Linear Multilinear Algebra. 1993;34:353–364]. In this paper, we give a normal version of a rank-r perturbation result due to Rado and published by Perfect [Duke Math. J. 1955;22:305–311], which allow us to obtain new sufficient conditions for the NNIEP to have a solution. These new conditions significantly improve Xu’s conditions. We also apply our results to construct nonnegative matrices with arbitrarily prescribed elementary divisors.

On nonnegative matrices with prescribed eigenvalues and diagonal entries

On the realizability of the critical points of a realizable list

The nonnegative inverse eigenvalue problem (NIEP) consists of finding necessary and sufficient conditions for the existence of a nonnegative matrix with a given list of complex numbers as its spectrum. If the matrix is required to be Leslie (doubly Leslie), the problem is called the Leslie (doubly Leslie) nonnegative eigenvalue inverse problem. In this paper, necessary and/or sufficient conditions for the existence and construction of Leslie and doubly Leslie matrices with a given spectrum are considered.

On universal realizability of spectra

A map of sufficient conditions for the symmetric nonnegative inverse eigenvalue problem

For a complex matrix M, we denote by Sp(M) the spectrum of M and by |M| its absolute value, that is the matrix obtained from M by replacing each entry of M by its absolute value. Let A be a nonnegative real matrix, we call a signing of A every real matrix B such that |B|=A. In this paper, we characterize the set of all signings of A such that Sp(B)=αSp(A) where α is a complex unit number. Our motivation comes from some recent results about the relationship between the spectrum of a graph and the skew spectra of its orientations.

We present two versions of the same inequality, relating the maximal diagonal entry of a nonnegative matrix to its eigenvalues. We demonstrate a matrix factorization of a companion matrix, which leads to a solution of the nonnegative inverse eigenvalue problem (denoted the nniep) for 4×4 matrices of trace zero, and we give some sufficient conditions for a solution to the nniep for 5×5 matrices of trace zero. We also give a necessary condition on the eigenvalues of a 5×5 trace zero nonnegative matrix in lower Hessenberg form. Finally, we give a brief discussion of the nniep in restricted cases. Keywords: Nonnegative matricesMaximal diagonal entrynniepCompanion matricesAMS Subject Classification: 15A1815A2315A42

On Sufficient Conditions for the RNIEP and their Margins of Realizability

In this note we extend the necessary and sufficient conditions of Boyle–Handleman [M. Boyle and D. Handelman, The spectra of nonnegative matrices via symbolic dynamics, Ann. Math. 133 (1991), pp. 249–316] and Kim–Ormes–Roush [K.H. Kim, N.S. Ormes, and F.W. Roush, The spectra of nonnegative integer matrices via formal power series, J. Am. Math. Soc. 13 (2000), pp. 773–806] for a nonzero eigenvalue multiset of primitive matrices over ℝ+ and Z +, respectively, to irreducible matrices.

In this article, the index of imprimitivity of an irreducible nonnegative matrix in the famous Perron-Frobenius theorem is studied within a more general framework, both in a more general tensor setting and in a more natural spectral symmetry perspective. A $k$-th order tensor has symmetric spectrum if the set of eigenvalues is symmetric under a group action with the group being a subgroup of the multiplicative group of $k$-th roots of unity.A sufficient condition, in terms of linear equations over the quotient ring, for a tensor possessing symmetric spectrum is given, which becomes also necessary when the tensor is nonnegative, symmetric and weakly irreducible, or an irreducible nonnegative matrix. Moreover, it is shown that for a weakly irreducible nonnegative tensor, the spectral symmetries are the same when either counting or ignoring multiplicities of the eigenvalues.In particular, the spectral symmetry (index of imprimitivity) of an irreducible nonnegative Sylvester matrix is completely resolved via characterizations with the indices of its positive entries. It is shown that the spectrum of an irreducible nonnegative Sylvester matrix can only be $1$-symmetric or $2$-symmetric, and the exact situations are fully described. With this at hand, the spectral symmetry of a nonnegative two-dimensional symmetric tensor with arbitrary order is also completely characterized.

Measurable eigenvectors for Hermitian matrix-valued polynomials

Let Λ=λ1…λn be a list of complex numbers. Λ is said to be realizable if there is a nonnegative matrix with spectrum Λ. The list Λ is said to be universally realizable UR if it is realizable for each possible Jordan canonical form JCF allowed by Λ. The problem of determining the universal realizability of Λ is called universal realizability problem URP. The first results concerning URP (formerly called nonnegative inverse elementary divisors problem) are due to H. Minc and they establish that if Λ is the spectrum of a diagonalizable positive matrix, then Λ is UR. In this chapter, we introduce new results that contain extensions of Minc’s results and that allow us to show the universal realizability of lists of complex numbers not positively realizable. We also prove new universal realizability criteria and structured universal realizability criteria.