scholarly journals On the Inverse Eigenvalue Problem for Irreducible Doubly Stochastic Matrices of Small Orders

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Quanbing Zhang ◽  
Changqing Xu ◽  
Shangjun Yang
Keyword(s):  
Eigenvalue Problem ◽  
Matrix Theory ◽  
Stochastic Matrix ◽  
Sufficient Conditions ◽  
Difficult Problem ◽  
Inverse Eigenvalue Problem ◽  
Complex Case ◽  
Real Spectrum ◽  
Complex Spectrum ◽  
Inverse Eigenvalue

The inverse eigenvalue problem is a classical and difficult problem in matrix theory. In the case of real spectrum, we first present some sufficient conditions of a realr-tuple (forr=2; 3; 4; 5) to be realized by a symmetric stochastic matrix. Part of these conditions is also extended to the complex case in the case of complex spectrum where the realization matrix may not necessarily be symmetry. The main approach throughout the paper in our discussion is the specific construction of realization matrices and the recursion when the targetedr-tuple is updated to a(r+1)-tuple.

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 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].

scholarly journals An Inverse Eigenvalue Problem for Jacobi Matrices

2011 ◽  
Vol 2011 ◽  
pp. 1-11
Author(s):  
Zhengsheng Wang ◽  
Baojiang Zhong
Keyword(s):  
Eigenvalue Problem ◽  
Numerical Algorithm ◽  
Jacobi Matrix ◽  
Sufficient Conditions ◽  
Inverse Eigenvalue Problem ◽  
Jacobi Matrices ◽  
Inverse Eigenvalue

A kind of inverse eigenvalue problem is proposed which is the reconstruction of a Jacobi matrix by given four or five eigenvalues and corresponding eigenvectors. The solvability of the problem is discussed, and some sufficient conditions for existence of the solution of this problem are proposed. Furthermore, a numerical algorithm and two examples are presented.

scholarly journals The inverse eigenvalue problem for Leslie matrices

2019 ◽  
Vol 35 ◽  
pp. 319-330 ◽  
Author(s):  
Luca Benvenuti
Keyword(s):  
Eigenvalue Problem ◽  
Sufficient Conditions ◽  
Nonnegative Matrix ◽  
Inverse Eigenvalue Problem ◽  
Leslie Matrix ◽  
Complex Numbers ◽  
Nonnegative Matrices ◽  
Minimal Dimension ◽  
Leslie Matrices ◽  
Inverse Eigenvalue

The Nonnegative Inverse Eigenvalue Problem (NIEP) is the problem of determining necessary and sufficient conditions for a list of $n$ complex numbers to be the spectrum of an entry--wise nonnegative matrix of dimension $n$. This is a very difficult and long standing problem and has been solved only for $n\leq 4$. In this paper, the NIEP for a particular class of nonnegative matrices, namely Leslie matrices, is considered. Leslie matrices are nonnegative matrices, with a special zero--pattern, arising in the Leslie model, one of the best known and widely used models to describe the growth of populations. The lists of nonzero complex numbers that are subsets of the spectra of Leslie matrices are fully characterized. Moreover, the minimal dimension of a Leslie matrix having a given list of three numbers among its spectrum is provided. This result is partially extended to the case of lists of $n > 2$ real numbers.

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.

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