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

About the persymmetric nonnegative inverse eigenvalue problem in low dimension

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 the realizability of the critical points of a realizable list

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

In this paper, the symmetric positive semidefinite Procrustes problem is considered. By using matrix inner product and matrix decomposition theory, an explicit expression of the solution is given. Based on the explicit expression given in this paper, it is easy to derive the explicit expression of the solution given by Woodgate [K.G. Woodgate. Least-squares solution of F = PG over positive semidefinite symmetric P . Linear Algebra Appl., 245:171–190, 1996.] and by Liao [A.P. Liao. On the least squares problem of a matrix equation. J. Comput. Math., 17:589–594, 1999.] for the Procrustes problem in some special cases. The explicit expression given in this paper also shows that the solution of the special inverse eigenvalue problem considered by Zhang [L. Zhang. A class of inverse eigenvalue problem for symmetric nonnegative definite matrices. J. Hunan Educational Inst., 2:11–17, 1995 (in Chinese).] can be computed exactly. Examples to illustrate the correctness of the theory results are given.

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

The inverse matrix eigenvalue problem is the problem of reconstruction of a matrix (determination of the unknown parameters of a certain matrix) for a given finite set of spectral data (points of the spectrum). The spectral data used in this case may contain complete or only partial information on the eigenvalues or eigenvectors. The aim of an inverse eigenvalue problem is to construct a matrix that preserves a certain specific structure and the prescribed spectral properties. There are two basic questions connected with any inverse eigenvalue problem: the theoretical question on the possibility of construction of its solutions and the practical question of its determination. The main investigations connected with solvability deal with establishing necessary or sufficient conditions under which the inverse eigenvalue problem is solvable. On the other hand, the main difficulty in calculations is connected with the development of a numerical procedure which would enable us to construct the matrix itself according to the a-priori specified spectral data. Both questions are important and complicated. Numerous works are devoted to the conditions of existence and uniqueness of solutions for different formulations of inverse matrix spectral problems and the methods used for their solution (see, e.g., [1–3, 7, 9–13, 15] and the literature therein). In the present work, we consider a numerical algorithm used for the solution of inverse eigenvalue problems under the assumption that this solution exists. Hence, we consider inverse algebraic eigenvalue problems.

The inverse eigenvalue problem of structured matrices from the design of Hopfield neural networks

This article considers an inverse eigenvalue problem for bisymmetric matrices under a central principal submatrix constraint and the corresponding optimal approximation problem. We first discuss the specified structure of bisymmetric matrices and their central principal submatrices. Then we study a special form for the matrix of independent eigenvectors for a bisymmetric matrix. Based on these, we give some necessary and sufficient conditions for the solvability of the inverse eigenvalue problem, and we derive an expression for its general solution. Finally, we obtain an expression for the solution to the corresponding optimal approximation problem.

Algorithm for obtaining characteristic polynomial (CP) coefficients of an alternant edge-weighted cycle is used to arrive at the algorithm for that of the cycloparaphenylene (CPP) graphs in matrix product form. The algorithm gives a recursive relation in expressing the sum of the CP coefficients of a CPP in terms of that of its two immediately preceding analogues which ultimately ends up with the use of transfer matrix in an analytical form. The sum of CP coefficients, being combinatorial in nature, is found to be used as a topological index showing much similarity with Hosoya index (sum all matching polynomial coefficients), cardinality and number of Kekulé valence structures of CPP graphs compared to the Wiener index which is the distance sum of all pairs of vertices in the graph. The sum of CP coefficients has been found to model the physical properties like strain energy and diameter of CPPs that are verified by the respective excellent correlations.

Summary Inverse eigenvalue problems associated with self-adjoint differential equations of order two or higher are considered. The question of what kind of data are necessary and sufficient to insure the existence of a unique solution is examined. A method of solution for well-posed inverse eigenvalue problems is then presented. The implications of this analysis to the inverse problem for the Earth's spheroidal modes of vibration is also discussed. In this talk, I propose to present a partial survey of the mathematical literature on inverse eigenvalue problems, with an eye toward geophysical applications, and to state some new results which I obtained recently. Perhaps the inverse eigenvalue problem which is best known to geophysicists is that related to the normal modes of vibration of the Earth. It can be stated thus: assuming that the Earth is spherically symmetric and given the natural frequencies ,S, and ,T, of the spheroidal and torsional modes of vibration, can we deduce the density p(r) and the speed of propagation Vr Takeuchi & Saito 1972) and very little is known about inverse eigenvalue problems of order higher than two. In order to gain an understanding of inverse eigenvalue problems one should perhaps divorce oneself from a specific problem and investigate rather a sequence of problems of increasing complexity. If you allow me to adopt this attitude, then the first task would be a rough classification of eigenvalue problems. Without pretending to be exhaustive, let me divide eigenvalue problems into two classes depending upon whether they are associated with ordinary or partial differential equations, i.e. depending upon whether they are related to one-dimensional or multi-dimensional problems. The following Sturm-Liouville problem: u+{n-q(x)}u = 0

The inverse eigenvalue problem for symmetric matrices (IEP) is formulated as a system of two matrix equations. For solving the system a variation of Newton's method is used which has been proposed by Fusco and Zecca [Calcolo XXIII (1986), pp. 285–303] for the simultaneous computation of eigenvalues and eigenvectors of a given symmetric matrix. an iteration step of this method consists of a Newton step followed by an orthonormalization with the consequence that each iterate satisfies one of the given equations. The method converges locally y quadratically to regular solutions. The algorithm and some numerical examples are presented. In addition, it is shown that the so called Method III proposed by Friedland, Nocedal, and Overton [SIAM J. Numer. Anal.,24 (1987), pp. 634–667] for solving IEP may be constructed similarly to the method presented here