scholarly journals Spectra inhabiting the left half-plane that are universally realizable

2021 ◽  
Vol 10 (1) ◽  
pp. 180-192
Author(s):  
Ricardo L. Soto
Keyword(s):  
Canonical Form ◽  
Half Plane ◽  
New Left ◽  
Nonnegative Matrix ◽  
Complex Numbers ◽  
Jordan Canonical Form ◽  
Positivity Condition ◽  
Left Half ◽  
Left Half Plane ◽  
New Criteria

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.

scholarly journals Permutative universal realizability

2021 ◽  
Vol 9 (1) ◽  
pp. 66-77
Author(s):  
Ricardo L. Soto ◽  
Ana I. Julio ◽  
Jaime H. Alfaro
Keyword(s):  
Canonical Form ◽  
Nonnegative Matrix ◽  
Positive Element ◽  
Complex Numbers ◽  
Sufficient Condition ◽  
Real Numbers ◽  
Jordan Canonical Form

Abstract A list of complex numbers Λ is said to be realizable, if it is the spectrum of a nonnegative matrix. In this paper we provide a new sufficient condition for a given list Λ to be universally realizable (UR), that is, realizable for each possible Jordan canonical form allowed by Λ. Furthermore, the resulting matrix (that is explicity provided) is permutative, meaning that each of its rows is a permutation of the first row. In particular, we show that a real Suleĭmanova spectrum, that is, a list of real numbers having exactly one positive element, is UR by a permutative matrix.

scholarly journals Centrosymmetric universal realizability

2021 ◽  
Vol 37 ◽  
pp. 680-691
Author(s):  
Ana Julio ◽  
Yankis R. Linares ◽  
Ricardo L. Soto
Keyword(s):  
Canonical Form ◽  
Sufficient Conditions ◽  
Nonnegative Matrix ◽  
Complex Numbers ◽  
Jordan Canonical Form ◽  
Centrosymmetric Matrix

A list $\Lambda =\{\lambda_{1},\ldots,\lambda_{n}\}$ of complex numbers is said to be realizable, if it is the spectrum of an entrywise nonnegative matrix $A$. In this case, $A$ is said to be a realizing matrix. $\Lambda$ is said to be universally realizable, if it is realizable for each possible Jordan canonical form (JCF) allowed by $\Lambda$. The problem of the universal realizability of spectra is called the universal realizability problem (URP). Here, we study the centrosymmetric URP, that is, the problem of finding a nonnegative centrosymmetric matrix for each JCF allowed by a given list $\Lambda $. In particular, sufficient conditions for the centrosymmetric URP to have a solution are generated.

scholarly journals The role of certain Brauer and Rado results in the nonnegative inverse spectral problems

2020 ◽  
Vol 36 (36) ◽  
pp. 484-502
Author(s):  
Ana Julio ◽  
Ricardo Soto
Keyword(s):  
Canonical Form ◽  
Sufficient Conditions ◽  
Nonnegative Matrix ◽  
Complex Numbers ◽  
Inverse Spectral Problems ◽  
Jordan Canonical Form ◽  
Spectral Problems ◽  
Inverse Spectral

It is said that a list $\Lambda =\{\lambda _{1},\ldots ,\lambda _{n}\}$ of complex numbers is realizable, if it is the spectrum of a nonnegative matrix $A$. It is said that $\Lambda $ is universally realizable if it is realizable for each possible Jordan canonical form allowed by $\Lambda$. This work does not contain new results. As its title says, its goal is to show and emphasize the relevance and importance of certain results, by Brauer and Rado, in the study of nonnegative inverse spectral problems. It is shown that virtually all known results, which give sufficient conditions for $\Lambda$ to be realizable or universally realizable, can be obtained from results by Brauer and Rado. Moreover, from these results, a realizing matrix may always be constructed.

scholarly journals Toeplitz nonnegative realization of spectra via companion matrices

2019 ◽  
Vol 7 (1) ◽  
pp. 230-245
Author(s):  
Macarena Collao ◽  
Mario Salas ◽  
Ricardo L. Soto
Keyword(s):  
Eigenvalue Problem ◽  
Half Plane ◽  
Toeplitz Matrix ◽  
Toeplitz Matrices ◽  
Nonnegative Matrix ◽  
Inverse Eigenvalue Problem ◽  
Complex Numbers ◽  
Inverse Eigenvalue ◽  
Companion Matrices ◽  
Solution Matrix

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 = {λ1, . . ., 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= {λ1, . . ., 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.

scholarly journals Conditions for the physical realizability of a typical component z(y)-matrix of a matching quadrupole of a general form in a concentrated elemental basis

2020 ◽  
pp. 13-20
Author(s):  
Gennady Devyatkov ◽  
Keyword(s):  
Half Plane ◽  
Single Point ◽  
Sufficient Conditions ◽  
Point Of View ◽  
Signal Source ◽  
Left Half ◽  
Left Half Plane ◽  
Broadband Matching ◽  
Typical Component ◽  
Physical Realizability

When solving problems of broadband matching, very often there is a need for a certain form of the amplitude-frequency characteristic. In connection with this, the problem comes up of synthesizing broadband matching devices that simultaneously have correcting properties, i.e. having a given frequency dependence of the power conversion coefficient in the operating frequency band. The use of broadband reactive matching - correcting circuits in most practical cases is difficult because of the reflected power. This leads to the problem of the synthesis of broadband matching-correcting circuits with arbitrary immittances of the signal source and load in an elemental basis of a general form, containing along with reactive and active elements, which has not been adequately solved. Therefore, it becomes necessary to find the conditions for the physical realizability of a typical component of the immitance matrix of a two-port network of general form containing poles in the left half-plane of complex frequencies. In this paper the necessary and sufficient conditions are defined for the physical realizability of the immitance matrix of a typical component of a subclass of two-terminal networks of general form in a lumped elemental electric basis, when the poles of the Eigen functions in the Foster representation can be in the left half-plane of complex frequencies, excluding the imaginary and real axes. This allows to synthesis of broadband dissipative matching, matching-correcting circuits and matched attenuators in an elemental basis of a general form with arbitrary immitances of the signal source and load from a single point of view.

scholarly journals Radii of starlikeness associated with the lemniscate of Bernoulli and the left-half plane

2012 ◽  
Vol 218 (11) ◽  
pp. 6557-6565 ◽  
Author(s):  
Rosihan M. Ali ◽  
Naveen K. Jain ◽  
V. Ravichandran
Keyword(s):  
Half Plane ◽  
Left Half ◽  
Lemniscate Of Bernoulli ◽  
Left Half Plane

scholarly journals Singular Values of One Parameter Family \(\lambda ((e^{z}-1)/z)^{m}\)

2015 ◽  
Vol 4 (2) ◽  
pp. 295 ◽  
Author(s):  
Mohammad Sajid
Keyword(s):  
Half Plane ◽  
Entire Functions ◽  
Singular Values ◽  
Critical Values ◽  
Parameter Family ◽  
Open Disk ◽  
Left Half ◽  
Left Half Plane

In the present paper, the singular values of one parameter family of entire functions $f_{\lambda}(z)=\lambda\bigg(\dfrac{e^{z}-1}{z}\bigg)^{m}$ and $f_{\lambda}(0)=\lambda$, $m\in \mathbb{N}\backslash \{0\}$, $\lambda\in \mathbb{R} \backslash \{0\}$, $z \in \mathbb{C}$ is investigated. It is shown that all the critical values of $f_{\lambda}(z)$ lie in the left half plane. It is also found that the function $f_{\lambda}(z)$ has infinitely many bounded singular values and lie inside the open disk centered at origin and having radius $|\lambda|$.

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