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The nonnegative inverse elementary divisors problem ( NIEDP ) is the problem of finding conditions for the existence of an n × n entrywise nonnegative matrix A with prescribed elementary divisors. We consider the case in which the solution matrix A is required to be persymmetric. Persymmetric matrices are common in physical sciences and engineering. They arise, for instance, in the control of mechanical and electric vibrations. In this paper, we solve the NIEDP for n × n matrices assuming that ( i ) there exists a partition of the given list Λ = { λ 1 , … , λ n } in sublists Λ k , along with suitably chosen Perron eigenvalues, which are realizable by nonnegative matrices A k with certain of the prescribed elementary divisors, and ( ii ) a nonnegative persymmetric matrix exists with diagonal entries being the Perron eigenvalues of the matrices A k , with certain of the prescribed elementary divisors. Our results generate an algorithmic procedure to compute the structured solution matrix. [ABSTRACT FROM AUTHOR]

Thenonnegative inverse eigenvalue problemis the problem of finding necessary and sufficient conditions for the existence of anentrywise nonnegative matrixAwith prescribed spectrum. This problem remains open for. If the matrixAis required to be normal, the problem will be called thenormal 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-rperturbation result due to Rado and published by Perfect [Duke Math. J. 1955;22:305–311], which allow us to obtain new sufficient conditions for theNNIEPto 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. [ABSTRACT FROM AUTHOR]

The nonnegative inverse eigenvalue problem ( NIEP ) is the problem of finding conditions for the existence of an n × n entrywise nonnegative matrix A with prescribed spectrum. This problem remains open for n ≥ 5 . If the matrix A is required to be persymmetric (bisymmetric), the problem will be called persymmetric ( bisymmetric ) nonnegative inverse eigenvalue problem ( PNIEP ) ( BNIEP ). Persymmetric and bisymmetric matrices are common in physical sciences and engineering. They arise, for instance, in the control of mechanical and electric vibrations. A persymmetric version of a perturbation result, due to Rado and presented by H. Perfect in [5] , is developed and used to give sufficient conditions for the PNIEP to have a solution. Our results generate an algorithmic procedure to compute the solution matrix. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we give new sufficient conditions for the existence and construction of nonnegative matrices with prescribed elementary divisors, which drastically improve and contain some of the previous known conditions. We also show how to perturb complex eigenvalues of a nonnegative matrix while keeping its nonnegativity. These results allow us, under certain conditions, to easily decide if a given list is realizable with prescribed elementary divisors. [Copyright &y& Elsevier]

SUMMARY A new realizability criterion for the real nonnegative inverse eigenvalue problemis introduced. This criterion is a nontrivial extension of a powerful previous sufficient condition, based on negativity compensation. If the criterion is satisfied, then we can always construct a realizing matrix. It is also proved that this new criterion is easily adaptable to be sufficient for the construction of a symmetric nonnegative matrixwith given spectrum. In a natural way, the criterion extends to a family of sufficient conditions for the problem to have a solution. Copyright © 2011 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Given a square matrix A, a Brauer's theorem [Brauer A., Limits for the characteristic roots of a matrix. IV. Applications to stochastic matrices, Duke Math. J., 1952, 19(1), 75-91] shows how to modify one single eigenvalue of A via a rank-one perturbation without changing any of the remaining eigenvalues. Older and newer results can be considered in the framework of the above theorem. In this paper, we present its application to stabilization of control systems, including the case when the system is noncontrollable. Other applications presented are related to the Jordan form of A and Wielandt's and Hotelling's deflations. An extension of the aforementioned Brauer's result, Rado's theorem, shows how to modify r eigenvalues of A at the same time via a rank- r perturbation without changing any of the remaining eigenvalues. The same results considered by blocks can be put into the block version framework of the above theorem. [ABSTRACT FROM AUTHOR]

Abstract: An outstanding result of Guo [W. Guo, Eigenvalues of nonnegative matrices, Linear Algebra Appl. 266 (1997) 261-270] establishes that if the list is the spectrum of an nonnegative matrix, where is its Perron eigenvalue and , then for any , the list is also the spectrum of a nonnegative matrix. In this paper we extend the result of Guo to elementary divisors. In particular, if is a nonnegative matrix with spectrum then, by means of two rank one perturbations, we construct a modified matrix , which is also nonnegative, with spectrum and we explicitly provide the Jordan canonical form of . [Copyright &y& Elsevier]

Abstract: Guo[W. Guo, Eigenvalues of nonnegative matrices, Linear Algebra Appl. 266 (1997) 261–270] sets the question: if the list is symmetrically realizable (that is, is the spectrum of a symmetric nonnegative matrix), and whether or not the list is also symmetrically realizable. In this paper we give an affirmative answer to this question in the case that the realizing matrix is circulant or left circulant. We also give a necessary and sufficient condition for to be the spectrum of a nonnegative left circulant matrix. [Copyright &y& Elsevier]

Abstract: We consider the following inverse eigenvalue problem: to construct a symmetrical tridiagonal matrix from the minimal and maximal eigenvalues of all its leading principal submatrices. We give a necessary and sufficient condition for the existence of such a matrix and for the existence of a nonnegative symmetrical tridiagonal matrix. Our results are constructive, in the sense that they generate an algorithmic procedure to construct the matrix. [Copyright &y& Elsevier]

Abstract: A result by Brauer, which shows how to modify one single eigenvalue of a matrix without changing any of the remaining eigenvalues, plays a relevant role in the study of the nonnegative inverse eigenvalue problem (NIEP). Perfect, in a long time ignored paper (1955) presents an extension of this result, which shows how to modify r eigenvalues of a matrix of order n, r < n, via a rank-r perturbation, without changing any of the n − r remaining eigenvalues. By using this extension, Perfect gives a realizability criterion for the real NIEP, which is not contained in Soto’s realizability criterion. In this work, by extending Perfect’s result, we give a new realizability criterion for the real NIEP, which contains Soto’s criterion. Thus, this new realizability criterion appears to be the most general sufficient condition for the real NIEP so far. We also contribute to the solution of the symmetric NIEP for n =5. [Copyright &y& Elsevier]

Abstract: Let Λ ={λ 1, λ 2,…, λ n } a set of real numbers. The real nonnegative inverse eigenvalue problem (RNIEP) is the problem of determining necessary and sufficient conditions in order that Λ be the spectrum of an entrywise nonnegative n × n matrix. If there exists a nonnegative matrix A with spectrum Λ we say that Λ is realized by A. Many realizability criteria for the existence of such a matrix A are known. This paper shows that a realizability criterion given by the author, which contains both Kellogg’s realizability criterion and Borobia’s realizability criterion, is sufficient for the existence of an n × n symmetric nonnegative matrix with prescribed spectrum Λ. [Copyright &y& Elsevier]

We uniquely and explicitly reconstruct the instantaneous intrinsic metric of the Kerr-Newman event horizon from the spectrum of its Laplacian. In the process we find that the angular momentum parameter, radius, area; and in the uncharged case, mass, can be written in terms of these eigenvalues. In the uncharged case this immediately leads to the unique and explicit determination of the Kerr metric in terms of the spectrum of the event horizon. Robinson’s “no hair” theorem now yields the corollary: One can “hear the shape” of noncharged stationary axially symmetric black hole space-times by listening to the vibrational frequencies of its event horizon only. [ABSTRACT FROM AUTHOR]

Abstract: Let be an unweighted rooted tree of k levels such that in each level the vertices have equal degree. Let d k−j+1 denotes the degree of the vertices in the level j. We find the eigenvalues of the adjacency matrix and of the Laplacian matrix of . They are the eigenvalues of principal submatrices of two nonnegative symmetric tridiagonal matrices of order k × k. The codiagonal entries for both matrices are , and , while the diagonal entries are zeros, in the case of the adjacency matrix, and d j , 1⩽ j ⩽k, in the case of the Laplacian matrix. Moreover, we give some results concerning to the multiplicity of the above mentioned eigenvalues. [Copyright &y& Elsevier]

Abstract: A result by Brauer shows how to modify one single eigenvalue of a matrix via a rank-one perturbation, without changing any of the remaining eigenvalues. This, together with the properties of real matrices with constant row sums, was exploited by the authors in a previous work in connection with the nonnegative inverse eigenvalue problem, obtaining conditions which are sufficient for the existence of an entrywise nonnegative matrix with prescribed spectrum. In this work we make use of Brauer’s Theorem again, to show that most of the previous results giving sufficient conditions for the real nonnegative inverse eigenvalue problem can be derived by using Brauer’s Theorem. Moreover, the technique is constructive, and there is an algorithmic procedure to construct a matrix realizing the spectrum. In particular, we show that if either Kellogg’s realizability criterion or Borobia’s realizability criterion is satisfied, then Soto’s realizability criterion is also satisfied. None of the converses are true. Thus, the condition given by Soto appears to be the most general sufficient condition so far for the real nonnegative inverse eigenvalue problem. [Copyright &y& Elsevier]

We consider the problem of the existence and construction of nonnegative matrices with prescribed spectrum and diagonal entries. Necessary and sufficient conditions have been obtained for n ≤ 3 , by Perfect and Fiedler, in the cases nonnegative and nonnegative symmetric, respectively. For n ≥ 4 , they obtained sufficient conditions. Many partial results about the problem have been published by several authors, mainly by Šmigoc. This is a long-standing unsolved inverse problem, but also necessary to apply a perturbation result, due to R. Rado, which has played an important role in the study of nonnegative inverse eigenvalue and inverse elementary divisors problems. Distinct versions of Rado's result have been also obtained for certain structured matrices. To apply Rado's result and its different versions we need to guarantee the existence of an r × r , r < n , nonnegative (structured) matrix with prescribed spectrum and diagonal entries. This is an important motivation for this work. Here, we prove new sufficient conditions for n ≥ 4 , which extend and strictly contain the Perfect and Fiedler conditions. Our results generate an algorithmic procedure to construct a solution matrix. [ABSTRACT FROM AUTHOR]

A list of eigenvalues is said to be realizable if it is the spectrum of a nonnegative matrix, diagonalizably realizable (DR) if it is the spectrum of a diagonalizable nonnegative matrix, and universally realizable (UR) if there is a nonnegative matrix with this spectrum and any possible Jordan canonical form allowed by the spectrum. The nonnegative inverse eigenvalue problem (NIEP) asks which lists are realizable. It is known that there exist spectra that are realizable, but not DR. We raise the question of whether DR implies UR. This is known in a few cases, including n ≤ 4 , nonnegative spectra, and Suleimanova spectra. We add some new classes of spectra that are UR. [ABSTRACT FROM AUTHOR]

Abstract: A connection is established between the problem of characterizing all possible real spectra of entrywise nonnegative matrices (the so-called real nonnegative inverse eigenvalue problem) and a combinatorial process consisting in repeated application of three basic manipulations on sets of real numbers. Given realizable sets (i.e., sets which are spectra of some nonnegative matrix), each of these three elementary transformations constructs a new realizable set. This defines a special kind of realizability, called C-realizability and this is closely related to the idea of compensation. After observing that the set of all C-realizable sets is a strict subset of the set of realizable ones, we show that it strictly includes, in particular, all sets satisfying several previously known sufficient realizability conditions in the literature. Furthermore, the proofs of these conditions become much simpler when approached from this new point of view. [Copyright &y& Elsevier]

Abstract: The following inverse eigenvalue problem was introduced and discussed in [J. Peng, X.Y. Hu, L. Zhang, Two inverse eigenvalue problems for a special kind of matrices, Linear Algebra Appl. 416 (2006) 336–347]: to construct a real symmetric bordered diagonal matrix A from the minimal and maximal eigenvalues of all its leading principal submatrices. However, the given formulae in [4, Theorem 1] to compute the matrix A may lead us to a matrix, which does not satisfy the requirements of the problem. In this paper, we rediscuss the problem to give a sufficient condition for the existence of such a matrix and necessary and sufficient conditions for the existence of a nonnegative such a matrix. Results are constructive and generate an algorithmic procedure to construct the matrices. [Copyright &y& Elsevier]

Abstract: The real nonnegative inverse eigenvalue problem (RNIEP) is the problem of determining necessary and sufficient conditions for a list of real numbers Λ to be the spectrum of an entrywise nonnegative matrix. A number of sufficient conditions for the existence of such a matrix are known. In this paper, in order to construct a map of sufficient conditions, we compare these conditions and establish inclusion relations or independency relations between them. [Copyright &y& Elsevier]

If a set Δ of complex numbers can be partitioned as Δ=Λ1∪⋯∪Λs in such a way that each Λi is realized as the spectrum of a nonnegative matrix, say Ai, then Δ is trivially realized as the spectrum of the nonnegative matrix A=⊕Ai. In [Linear Algebra Appl. 369 (2003) 169] it was shown that, in some cases, a real set Δ can be realized even if some of the Λi are not realizable themselves. Here we systematize and extend these results, in particular allowing the sets to be complex. The leading idea is that one can associate to any nonrealizable set Γ a certain negativity N(Γ), and to any realizable set Λ a certain positivity M(Λ). Then, under appropriate conditions, if M(Λ)⩾N(Γ) we can conclude that Γ∪Λ is the spectrum of a nonnegative matrix. Additionally, we prove a complex generalization of Suleimanova’s theorem. [Copyright &y& Elsevier]

We say that a list Λ = { λ 1 , ... , λ n } of complex numbers is realizable, if it is the spectrum of a nonnegative matrix A (a realizing matrix). We say that Λ is universally realizable if it is realizable for each possible Jordan canonical form allowed by Λ. This work studies the universal realizability of spectra in low dimension, that is, realizable spectra of size n ≤ 5. It is clear that for n ≤ 3 the concepts of universally realizable and realizable are equivalent. The case n = 4 is easily deduced from previous results in [7]. We characterize the universal realizability of real spectra of size 5 and trace zero, and we describe a region for the universal realizability of nonreal 5-spectra with trace zero. As an important by-product of our study, we also show that realizable lists on the left half-plane, that is, lists Λ = { λ 1 , ... , λ n } , where λ 1 is the Perron eigenvalue and Re λ i ≤ 0 , for i = 2 , ... , n , are not necessarily universally realizable. [ABSTRACT FROM AUTHOR]

Abstract A list Λ = { λ 1 , λ 2 , ... , λ n } of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix. The list Λ is said to be universally realizable (UR) if it is the spectrum of a nonnegative matrix for each possible Jordan canonical form allowed by Λ. It is well known that an n × n nonnegative matrix A is co-spectral to a nonnegative matrix B with constant row sums. In this paper, we extend the co-spectrality between A and B to a similarity between A and B , when the Perron eigenvalue is simple. We also show that if ϵ ≥ 0 and Λ = { λ 1 , λ 2 , ... , λ n } is UR , then { λ 1 + ϵ , λ 2 , ... , λ n } is also UR. We give counter-examples for the cases: Λ = { λ 1 , λ 2 , ... , λ n } is UR implies { λ 1 + ϵ , λ 2 − ϵ , λ 3 , ... , λ n } is UR , and Λ 1 , Λ 2 are UR implies Λ 1 ∪ Λ 2 is UR. [ABSTRACT FROM AUTHOR]