Nonnegative Realization Research Articles

On new universal realizability criteria

Abstract A list Λ = { λ 1 , λ 2 , … , λ n } \Lambda =\left\{{\lambda }_{1},{\lambda }_{2},\ldots ,{\lambda }_{n}\right\} of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix and is said to be universally realizable (UR), if it is realizable for each possible Jordan canonical form allowed by Λ \Lambda . In 1981, Minc proved that if Λ \Lambda is diagonalizably positively realizable, then Λ \Lambda is UR [Proc. Amer. Math. Society 83 (1981), 665–669]. The question whether this result holds for nonnegative realizations was open for almost 40 years. Recently, two extensions of Mins’s result have been obtained by Soto et al. [Spec. Matrices 6 (2018), 301–309], [Linear Algebra Appl. 587 (2020), 302–313]. In this work, we exploit these extensions to generate new universal realizability criteria. Moreover, we also prove that under certain conditions, the union of two lists UR is also UR, and for certain criteria, if Λ \Lambda is UR, then for t ≥ 0 t\ge 0 , Λ t = { λ 1 + t , λ 2 ± t , λ 3 , … , λ n } {\Lambda }_{t}=\left\{{\lambda }_{1}+t,{\lambda }_{2}\pm t,{\lambda }_{3},\ldots ,{\lambda }_{n}\right\} is also UR.

On the doubly stochastic realization of spectra

An n-list $$\lambda = \left( r; \lambda _2, \ldots , \lambda _n\right)$$ of complex numbers with $$r>\max _{2\le j\le n}|\lambda _j|,$$ is said to be realizable if $$\lambda$$ is the spectrum of $$n\times n$$ nonnegative matrix A and in this case A is said to be a nonnegative realization of $$\lambda .$$ If, in addition, each row and column sum of A is equal to r, then $$\lambda$$ is said to be doubly stochastically realizable and in such case A is said to be a doubly stochastic realization for $$\lambda .$$ In 1997, Guo proved that if $$\left( \lambda _2,\ldots , \lambda _n\right)$$ is any list of complex numbers which is closed under complex conjugation then there exists a least real number $$\lambda _0$$ with $$\max _{2\le j\le n}|\lambda _j|\le \lambda _0\le 2n\max _{2\le j\le n}|\lambda _j|$$ such that the list of complex numbers $$\left( \lambda _1,\lambda _2,\ldots ,\lambda _n\right)$$ is realizable if and only if $$\lambda _1\ge \lambda _0.$$ Many researchers deal with sharpening this upper bound and others are concerned with finding the exact value of $$\lambda _0$$ for particular classes of matrices (Andrade et al. in Linear Algebra Appl 556:301–322, 2018; Andrade et al. in Linear Algebra Appl 551:36–56, 2018; Julio and Soto in Electron J Linear Algebra 36:484–502, 2020; Robbiano in Linear Algebra Appl 564:15–27, 2019). In this paper, we first describe a method for passing from a nonnegative realization to a doubly stochastic realization. As applications, we give a new sufficient condition for a stochastic matrix A to be cospectral to a doubly stochastic matrix B and in this case B is shown to be the unique closest doubly stochastic matrix to A with respect to the Frobenius norm. Then, our next result gives an improvement of Guo’s bound which also sharpens the existing known bound for the case when one of at least one of $$\left\{ \lambda _2, \ldots , \lambda _n\right\}$$ is real. Furthermore, we investigate the case when $$\left\{ \lambda _2, \ldots , \lambda _n\right\}$$ are all non-real which has not been dealt before. Our main results here also sharpen Guo’s bound. Next, for doubly stochastic realizations, we obtain an upper bound that improves Guo’s bound as well. Finally, for certain particular cases, we give a further improvement of our last bound for doubly stochastic realization.

Spectra inhabiting the left half-plane that are universally realizable

Abstract Let Λ = {λ1, λ2, . . ., λ n } be a list of complex numbers. Λ is said to be realizable if it is the spectrum of an entrywise nonnegative matrix. Λ is universally realizable if it is realizable for each possible Jordan canonical form allowed by Λ. Minc ([21],1981) showed that if Λ is diagonalizably positively realizable, then Λ is universally realizable. The positivity condition is essential for the proof of Minc, and the question whether the result holds for nonnegative realizations has been open for almost forty years. Recently, two extensions of the Minc’s result have been proved in ([5], 2018) and ([12], 2020). In this work we characterize new left half-plane lists (λ1 > 0, Re λ i ≤ 0, i = 2, . . ., n) no positively realizable, which are universally realizable. We also show new criteria which allow to decide about the universal realizability of more general lists, extending in this way some previous results.

Centrosymmetric nonnegative realization of spectra

A list Λ={λ1,λ2,…,λn} of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix. In this paper we intent to characterize those lists of complex numbers, which are realizable by a centrosymmetric nonnegative matrix. In particular, we show that lists of nonnegative real numbers, and lists of complex numbers of Suleimanova type (except in one particular case), are always the spectrum of some centrosymmetric nonnegative matrix. For the general lists we give sufficient conditions via a perturbation result. We also show that for n=4, every realizable list of real numbers is also realizable by a nonnegative centrosymmetric matrix.

Toeplitz nonnegative realization of spectra via companion matrices

Abstract 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 Λ = {λ 1, . . ., λn}. If the problem has a solution, we say that Λ is realizable and that A is a realizing matrix. In this paper we consider the NIEP for a Toeplitz realizing matrix A, and as far as we know, this is the first work which addresses the Toeplitz nonnegative realization of spectra. We show that nonnegative companion matrices are similar to nonnegative Toeplitz ones. We note that, as a consequence, a realizable list Λ= {λ 1, . . ., λn} of complex numbers in the left-half plane, that is, with Re λi ≤ 0, i = 2, . . ., n, is in particular realizable by a Toeplitz matrix. Moreover, we show how to construct symmetric nonnegative block Toeplitz matrices with prescribed spectrum and we explore the universal realizability of lists, which are realizable by this kind of matrices. We also propose a Matlab Toeplitz routine to compute a Toeplitz solution matrix.

Normal nonnegative realization of spectra

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 realization of partitioned spectra

We consider partitioned lists of real numbers � = {�1,�2,...,�n}, and give efficient and constructive sufficient conditions for the existence of nonnegative and symmetric nonnegative matrices with spectrum �. Our results extend the ones given in (R.L. Soto and O. Rojo. Applications of a Brauer theorem in the nonnegative inverse eigenvalue problem. Linear Algebra Appl., 416:844- 856, 2006.) and (R.L. Soto, O. Rojo, J. Moro, and A. Borobia. Symmetric nonnegative realization of spectra. Electron. J. Linear Algebra, 16:1-18, 2007.) for the real and symmetric nonnegative inverse eigenvalue problem. We also consider the complex case and show how to construct an r × r nonnegative matrix with prescribed complex eigenvalues and diagonal entries.

Nonnegatively realizable spectra with two positive eigenvalues

Boyle and Handelman [M. Boyle and D. Handelman, The spectra of nonnegative matrices via symbolic dynamics, Ann. Math. 133 (1991), pp. 249–316.] characterized all lists of n complex numbers that can be the nonzero spectrum of a nonnegative matrix. This article presents a constructive proof of this result in the special case when the lists are real and contain two positive numbers and n − 2 negative numbers. A bound for the number of zeros that needs to be added to the list to achieve a nonnegative realization is presented in this case.

Nonnegative realization of complex spectra

We consider a list of complex numbers Λ = {λ1,λ2,.. .,λn} and give a simple and efficient sufficient condition for the existence of an n × n nonnegative matrix with spectrum Λ. Our result extends a previous one for a list of real numbers given in [Linear Algebra Appl., 416:844–856, 2006]. In particular, we show how to construct a nonnegative matrix with prescribed complex eigenvalues and diagonal entries. As a by-product, we also construct Hermitian matrices with prescribed spectrum, whose entries have nonnegative real parts.

Symmetric nonnegative realization of spectra

A perturbation result, due to R. Rado and presented by H. Perfect in 1955, shows how to modify r eigenvalues of a matrix of order n, r ≤ n, via a perturbation of rank r, without changing any of the n − r remaining eigenvalues. This result extended a previous one, due to Brauer, on perturbations of rank r = 1. Both results have been exploited in connection with the nonnegative inverse eigenvalue problem. In this paper a symmetric version of Rado’s extension is given, which allows us to obtain a new, more general, sufficient condition for the existence of symmetric nonnegative matrices with prescribed spectrum.

Nonnegative realization of spectra having negative real parts

The nonnegative inverse eigenvalue problem (NIEP) is the problem of determining necessary and sufficient conditions for a list of complex numbers σ to be the spectrum of a nonnegative matrix. In this paper the problem is completely solved in the case when all numbers in the given list σ except for one (the Perron eigenvalue) have real parts smaller than or equal to zero.

Driving variable realizations and the non-negative realization problem for controllable behaviours

Non-negative linear systems, traditionally investigated within the state-space framework, have been recently analysed within the behavioural setting. In a couple of recent papers [J. W. Nieuwenhuis, “When to call a linear system nonnegative”, Linear Algebra & its Appl., 281, 1998, pp. 43–58.; M. E. Valcher, “Nonnegative linear systems in the behavioural approach: the autonomous case”, Linear Algebra and its Appl., 2000, 319, pp. 147–162.], several definitions and results about non-negative behaviours (and, in particular, about non-negative autonomous behaviours) have been derived. Moreover, the non-negative realization problem for autonomous behaviours has been fully explored in [M. E. Valcher, “Non-negative realization of autonomous systems in the behavioral approach”, SIAM J. on Control and Opt., 2001, 40, pp. 540–556.], thus deriving an extended set of necessary and sufficient conditions for an autonomous behaviour to admit a non-negative realization. Here, we focus our attention on the non-negative realization problem for controllable behaviours. To this end, we first address the general realization problem by means of a driving variable (DV) state-space representation, and investigate under what conditions a state-space model provides a DV realization of a controllable behaviour. Based on these results, we analyse the possibility of realizing a controllable behaviour by means of a non-negative DV representation and, in particular, a reachable non-negative DV realization. Moreover, necessary and sufficient conditions for a behaviour to be the controllable part of a (complete) behaviour endowed with a non-negative realization are presented.

Connections between Realizations of a Transfer Function

It is one of the basic facts of linear time-invariant systems theory that any two minimal (canonical) realizations are connected in the best possible way: by system similarity. We study five different types of possible connections between two arbitrary realizations of a transfer function, and are interested in questions of existence (sufficient and/or necessary conditions), uniqueness, and description of all (or of a possibly large class of) connecting operators or pairs of operators. In the case of the existence of nonnegative realizations we seek nonnegative connecting pairs or operators.

Nonnegative Realization of Autonomous Systems in the Behavioral Approach

Nonnegative linear systems, which have traditionally been investigated within the state-space framework, have been recently introduced and analyzed by means of the behavioral approach. In a couple of recent papers, [J. W. Nieuwenhuis, Linear Algebra Appl., 281 (1998), pp. 43--58, M. E. Valcher, Linear Algebra Appl., 319 (2000), pp. 147--162], several general definitions and results about nonnegative behaviors, as well as a complete analysis of nonnegativity property for autonomous behaviors, have been presented. In this contribution, by focusing our interest again on autonomous behaviors, we explore the nonnegative realization problem by deriving an extended set of necessary and sufficient (geometric) conditions for an autonomous behavior to be nonnegative realizable. In the scalar case, in particular, necessary and sufficient conditions for nonnegative realizability, which refer to the set of zeros of any polynomial involved in the kernel description of the behavior, are provided. Finally, a comparison between the nonnegative realizability property, here investigated, and K-realizability, addressed in [H. Maeda and S. Kodama, IEEE Trans. ircuits, Systems I Fund. Theory Appl., CAS-281 (1981), pp. 39--47] is carried on.

Nonnegative realizations of matrix transfer functions

Various aspects of the nonnegative, finite-dimensional realizability of time-invariant discrete linear systems are considered. A new proof of the basic result of the nonnegative realizability of a primitive (scalar-valued) transfer function with nonnegative impulse response function is given. An algorithm for establishing whether a scalar-valued transfer function with nonnegative impulse response has a nonnegative realization is presented. The main result characterizes the nonnegative realizability of a scalar-valued transfer function with the help of primitive transfer functions, and is extended to the general case of matrix-valued transfer functions. Then conditions for the existence of some special nonnegative realizations of transfer functions are presented, e.g., where the middle (main) matrix is irreducible, strictly positive or primitive.

Bounds on the size of minimal nonnegative realizations for discrete-time LTI systems

Nonnegative linear time-invariant (LTI) state-space models are characterized by the property that their state variables and outputs remain nonnegative whenever the initial state and the inputs are nonnegative. Necessary and sufficient conditions for the existence of a nonnegative state-space realization of a given discrete-time (DT) impulse response were obtained recently. In this letter, we look at minimal nonnegative realizations of DT impulse responses and develop bounds on the number of state variables that are necessary.

Spectral properties of rational matrix functions with nonnegative realizations

If ( A,B,C) is an (entrywise) nonnegative realization of a rational matrix function W (i.e. W( λ) = C( λ − A) −1 B for λ ∉ σ( A)) vanishing at infinity, then r( W) := inf{ r ⩾ 0: W has no poles λ with r < | λ|} is a pole of W and r( A) := spectral radius of A is an eigenvalue of A. We prove that, if the realization is minimal-nonnegative, then 1. 1. r( W) = r( A), 2. 2. order of the pole r( W) of W = order of the pole r( A) of (· − A) −1. We characterize the order of these poles in the spirit of Rothblum's index theorem, namely as the length of the longest chains of singular vertices in the reduced graph of A with a suitable new access relation, which incorporates B and C into the familiar access relation of A.

Nonnegative realization of a linear system with nonnegative impulse response

Let H(z) be a rational transfer function, with associated nonnegative impulse response sequence. The paper considers the question: When does there exist a triple A/spl isin/R/sup N/spl times/N/, b/spl isin/R/sup N/, c/spl isin/R/sup N/ with all nonnegative entries H(z)=c'(zI-A)/sup -1/b? An essentially complete characterization is given of the H(z) allowing such a realization, in terms of the location of the pole or poles of H(z) with maximum modulus.

A note on discrete-time positive realizations

A discrete-time positive linear system has the property that any nonnegative initial state and any nonnegative input produces a nonnegative trajectory in the state space and output for all time. Such systems are R n+- reachable if every state of the nonnegative orthant R n+ is reachable from origin with a nonnegative input function, in a finite-time interval. A simple necessary and sufficient condition for R n+- realizability , i.e. for the existence of a nonnegative realization that is also R n+- reachable , is stated. The realization procedure is straightforward.

Nonnegative realizations of systems over nonnegative quasi-fields

In this paper, several sufficient conditions of nonnegative realizability of nonnegative systems have been derived. The authors have also pointed out the relation between two kinds of nonnegative realizability.