The symmetric nonnegative inverse eigenvalue problem is to determine when a multiset of n real numbers is the spectrum of a symmetric nonnegative
matrix. The problem is solved only for
. In this ...article, the authors examine the set of possible spectra for nonnegative symmetric
matrices, dividing the set into points that are realizable, points that are unrealizable by known necessary conditions, and points that satisfy all known necessary conditions, but their realizability remains unknown.
A list Λ={λ1,…,λn} of complex numbers is said to be realizable if there exists an n×n nonnegative matrix A whose spectrum is Λ. In this case A is called a realizing matrix for Λ. The problem of ...characterizing all realizable lists Λ is known as the Nonnegative Inverse Eigenvalue Problem (NIEP). If A is required to be persymmetric, the Persymmetric Nonnegative Inverse Eigenvalue Problem (PNIEP) arises. In this paper, we show that the NIEP and the PNIEP are equivalent for any realizable list of three complex numbers. Furthermore, we give a new sufficient condition for a list of three complex numbers to be the spectrum of a persymmetric nonnegative matrix with prescribed diagonal entries. For some realizable lists of four complex numbers, we show that the NIEP and the PNIEP are equivalent. In general, we show that these problems are different. Finally, we found a sufficient condition for a trace zero list of five complex numbers to be the spectrum of a persymmetric nonnegative matrix.
Given a graph G, consider the family of real symmetric matrices with the property that the pattern of their nonzero off-diagonal entries corresponds to the edges of G. For the past 30 years a central ...problem has been to determine which spectra are realizable in this matrix class. Using combinatorial methods, we identify a family of graphs and multiplicity lists whose realizable spectra are highly restricted. In particular, we construct trees with multiplicity lists that require a unique spectrum, up to shifting and scaling. This represents the most extreme possible failure of spectral arbitrariness for a multiplicity list, and greatly extends all previously known instances of this phenomenon, in which only single linear constraints on the eigenvalues were observed.
The parameter q(G) of a graph G is the minimum number of distinct eigenvalues over the family of symmetric matrices described by G. It is shown that the minimum number of edges necessary for a ...connected graph G to have q(G)=2 is 2n−4 if n is even, and 2n−3 if n is odd. In addition, a characterization of graphs for which equality is achieved in either case is given.
Abstract
This paper studies the inverse eigenvalue problem for an arrow-shaped generalised Jacobi matrix, inverting matrices through two eigen-pairs. In the paper, the existence and uniqueness of the ...solution to the problem are discussed, and mathematical expressions as well as a numerical example are given. Finally, the uniqueness theorem of its matrix is established by mathematical derivation.
This paper is concerned with the inverse eigenvalue problem (IEP) for “fixed–fixed” mass–spring–inerter systems. Unlike the “fixed–free” case, a “fixed–fixed” system has its both ends attached to the ...ground. This brings some essential differences in the IEP. We find that the construction strategy proposed in Liu et al. (2022) cannot be readily applied here. So we endeavor to develop a new strategy in this paper to solve the problem and accordingly derive a necessary and sufficient condition for the “fixed–fixed” case.
The symmetric stochastic inverse eigenvalue problem (SSIEP) asks which lists of real numbers occur as the spectra of symmetric stochastic matrices. When the cardinality of a list is 4, Kaddoura and ...Mourad provided a sufficient condition for SSIEP by a mapping and convexity technique. They also conjectured that the sufficient condition is the necessary condition. This study presents the same sufficient condition for SSIEP, but we do it in terms of the list elements. In this way, we provide a different but more straightforward construction of symmetric stochastic matrices for SSIEP compared to those of Kaddoura and Mourad.
The symmetric nonnegative inverse eigenvalue problem (SNIEP) is the problem of characterizing all possible spectra of entry-wise nonnegative symmetric matrices of given dimension. A list of real ...numbers is said to be symmetrically realizable if it is the spectrum of some nonnegative symmetric matrix. One of the most general sufficient conditions for realizability is the so-called C-realizability, which amounts to some kind of compensation between the positive and negative entries of the list of real numbers whose realizability one is trying to decide. A combinatorial characterization of C-realizable lists with zero trace was given in 11. In this paper we make use of a recursive method for constructing simmetrically realizable lists due to Ellard and Šmigoc 3 to extend this combinatorial characterization of C-realizability to general lists with nonnegative trace. One consequence of this characterization is that the set of nonnegative C-realizable lists is a union of polyhedral cones whose faces are described by equations involving only linear combinations with coefficients 1 and −1 of the entries in the list. Another remarkable consequence is the monotonicity of C-realizability, i.e., the operation of increasing any positive entry of a C-realizable list preserves C-realizability.
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