scholarly journals Updating a map of sufficient conditions for the real nonnegative inverse eigenvalue problem

2019 ◽  
Vol 7 (1) ◽  
pp. 246-256 ◽  
Author(s):  
C. Marijuán ◽  
M. Pisonero ◽  
Ricardo L. Soto
Keyword(s):  
Eigenvalue Problem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Inverse Eigenvalue Problem ◽  
Real Matrix ◽  
Real Numbers ◽  
The Real ◽  
Inclusion Relations ◽  
Inverse Eigenvalue ◽  
Necessary And Sufficient

Abstract The real nonnegative inverse eigenvalue problem (RNIEP) asks for necessary and sufficient conditions in order that a list of real numbers be the spectrum of a nonnegative real matrix. A number of sufficient conditions for the existence of such a matrix are known. The authors gave in [11] a map of sufficient conditions establishing inclusion relations or independency relations between them. Since then new sufficient conditions for the RNIEP have appeared. In this paper we complete and update the map given in [11].

The Inverse Eigenvalue Problem for Symmetric Doubly Arrow Matrices

2013 ◽  
Vol 444-445 ◽  
pp. 625-627
Author(s):  
Kan Ming Wang ◽  
Zhi Bing Liu ◽  
Xu Yun Fei
Keyword(s):  
Eigenvalue Problem ◽  
Special Kind ◽  
Inverse Eigenvalue Problem ◽  
Symmetric Matrices ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Inverse Eigenvalue ◽  
Necessary And Sufficient

In this paper we present a special kind of real symmetric matrices: the real symmetric doubly arrow matrices. That is, matrices which look like two arrow matrices, forward and backward, with heads against each other at the station, . We study a kind of inverse eigenvalue problem and give a necessary and sufficient condition for the existence of such matrices.

Submatrix Constrained Inverse Eigenvalue Problem involving Generalised Centrohermitian Matrices in Vibrating Structural Model Correction

2016 ◽  
Vol 6 (1) ◽  
pp. 42-59 ◽  
Author(s):  
Wei-Ru Xu ◽  
Guo-Liang Chen
Keyword(s):  
Eigenvalue Problem ◽  
Structural Model ◽  
Sufficient Conditions ◽  
Approximation Problem ◽  
Inverse Eigenvalue Problem ◽  
Frobenius Norm ◽  
Structured Matrix ◽  
Inverse Eigenvalue ◽  
Left And Right ◽  
Necessary And Sufficient

AbstractGeneralised centrohermitian and skew-centrohermitian matrices arise in a variety of applications in different fields. Based on the vibrating structure equation where M, D, G, K are given matrices with appropriate sizes and x is a column vector, we design a new vibrating structure mode. This mode can be discretised as the left and right inverse eigenvalue problem of a certain structured matrix. When the structured matrix is generalised centrohermitian, we discuss its left and right inverse eigenvalue problem with a submatrix constraint, and then get necessary and sufficient conditions such that the problem is solvable. A general representation of the solutions is presented, and an analytical expression for the solution of the optimal approximation problem in the Frobenius norm is obtained. Finally, the corresponding algorithm to compute the unique optimal approximate solution is presented, and we provide an illustrative numerical example.

scholarly journals Brauer's theorem and nonnegative matrices with prescribed diagonal entries

2019 ◽  
Vol 35 ◽  
pp. 53-64 ◽  
Author(s):  
Ricardo Soto ◽  
Ana Julio ◽  
Macarena Collao
Keyword(s):  
Inverse Problem ◽  
Eigenvalue Problem ◽  
Inverse Eigenvalue Problem ◽  
Complex Numbers ◽  
Nonnegative Matrices ◽  
Simple Condition ◽  
Inverse Spectral Problems ◽  
Real Numbers ◽  
Inverse Eigenvalue ◽  
Necessary And Sufficient

The problem of the existence and construction of nonnegative matrices with prescribed eigenvalues and diagonal entries is an important inverse problem, interesting by itself, but also necessary to apply a perturbation result, which has played an important role in the study of certain nonnegative inverse spectral problems. A number of partial results about the problem have been published by several authors, mainly by H. \v{S}migoc. In this paper, the relevance of a Brauer's result, and its implication for the nonnegative inverse eigenvalue problem with prescribed diagonal entries is emphasized. As a consequence, given a list of complex numbers of \v{S}migoc type, or a list $\Lambda = \left\{\lambda _{1},\ldots ,\lambda _{n} \right \}$ with $\operatorname{Re}\lambda _{i}\leq 0,$ $\lambda _{1}\geq -\sum\limits_{i=2}^{n}\lambda _{i}$, and $\left\{-\sum\limits_{i=2}^{n}\lambda _{i},\lambda _{2},\ldots ,\lambda _{n} \right\}$ being realizable; and given a list of nonnegative real numbers $% \Gamma = \left\{\gamma _{1},\ldots ,\gamma _{n} \right\}$, the remarkably simple condition $\gamma _{1}+\cdots +\gamma _{n} = \lambda _{1}+\cdots +\lambda _{n}$ is necessary and sufficient for the existence and construction of a realizing matrix with diagonal entries $\Gamma .$ Conditions for more general lists of complex numbers are also given.

scholarly journals On the Solutions of Some Linear Complex Quaternionic Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Cennet Bolat ◽  
Ahmet İpek
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sylvester Equation ◽  
Real Matrix ◽  
Matrix Representations ◽  
Linear Complex ◽  
The Real ◽  
Special Cases ◽  
Necessary And Sufficient

Some complex quaternionic equations in the typeAX-XB=Care investigated. For convenience, these equations were called generalized Sylvester-quaternion equations, which include the Sylvester equation as special cases. By the real matrix representations of complex quaternions, the necessary and sufficient conditions for the solvability and the general expressions of the solutions are obtained.

A note on the real nonnegative inverse eigenvalue problem

2016 ◽  
Vol 31 ◽  
pp. 765-773 ◽  
Author(s):  
Raphael Loewy
Keyword(s):  
Eigenvalue Problem ◽  
Special Form ◽  
Nonnegative Matrix ◽  
Inverse Eigenvalue Problem ◽  
Real Numbers ◽  
The Real ◽  
Direct Sum ◽  
Inverse Eigenvalue

The Real Nonnegative Inverse Eigenvalue Problem (RNIEP) asks when is a list \[ \sigma=(\lambda_1, \lambda_2,\ldots,\lambda_n)\] consisting of real numbers the spectrum of an $n \times n$ nonnegative matrix $A$. In that case, $\sigma$ is said to be realizable and $A$ is a realizing matrix. In a recent paper dealing with RNIEP, P.~Paparella considered cases of realizable spectra where a realizing matrix can be taken to have a special form, more precisely such that the entries of each row are obtained by permuting the entries of the first row. A matrix of this form is called permutative. Paparella raised the question whether any realizable list $\sigma$ can be realized by a permutative matrix or a direct sum of permutative matrices. In this paper, it is shown that in general the answer is no.

scholarly journals An Inverse Eigenvalue Problem for Damped Gyroscopic Second-Order Systems

2009 ◽  
Vol 2009 ◽  
pp. 1-10 ◽  
Author(s):  
Yongxin Yuan
Keyword(s):  
Eigenvalue Problem ◽  
Symmetric Matrix ◽  
Sufficient Conditions ◽  
Positive Semidefinite ◽  
Inverse Eigenvalue Problem ◽  
Stiffness Matrices ◽  
Inverse Eigenvalue ◽  
Second Order Systems ◽  
Necessary And Sufficient ◽  
Semidefinite Matrix

The inverse eigenvalue problem of constructing symmetric positive semidefinite matrix (written as ) and real-valued skew-symmetric matrix (i.e., ) of order for the quadratic pencil , where , are given analytical mass and stiffness matrices, so that has a prescribed subset of eigenvalues and eigenvectors, is considered. Necessary and sufficient conditions under which this quadratic inverse eigenvalue problem is solvable are specified.

Characterizations of symmetric cones by means of the basic relative invariants of homogeneous cones

2016 ◽  
Vol 7 (2) ◽  
Author(s):  
Hideto Nakashima
Keyword(s):  
Sufficient Conditions ◽  
Rational Functions ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Symmetric Cones ◽  
The Real ◽  
Tube Domains ◽  
Homogeneous Cone ◽  
Necessary And Sufficient ◽  
Relative Invariants

AbstractIn this paper, we give necessary and sufficient conditions for a homogeneous cone Ω to be symmetric in two ways. One is by using the multiplier matrix of Ω, and the other is in terms of the basic relative invariants of Ω. In the latter approach, we need to show that the real parts of certain meromorphic rational functions obtained by the basic relative invariants are always positive on the tube domains over Ω. This is a generalization of a result of Ishi and Nomura [Math. Z. 259 (2008), 604–674].

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