Inverse eigenvalue problem for Jacobi matrix and nonnegative matrices

2021 ◽  
pp. 2250098
Author(s):  
Somayeh Zangoei Zadeh ◽  
Azim Rivaz
Keyword(s):  
Eigenvalue Problem ◽  
Jacobi Matrix ◽  
Inverse Eigenvalue Problem ◽  
Nonnegative Matrices ◽  
Inverse Eigenvalue ◽  
Matrix Formula

In this paper, we present a method for constructing a Jacobi matrix [Formula: see text] using [Formula: see text] known eigenvalues [Formula: see text]. Some conditions are also given under which the constructed matrix is nonnegative and its diagonal entries are specified. Finally, we present a technique for constructing symmetric and nonsymmetric nonnegative matrices by their eigenvalues.

scholarly journals On the inverse eigenvalue problem of symmetric nonnegative matrices

2019 ◽  
Vol 14 (1) ◽  
pp. 11-19
Author(s):  
A. M. Nazari ◽  
A. Mashayekhi ◽  
A. Nezami
Keyword(s):  
Eigenvalue Problem ◽  
Nonnegative Matrix ◽  
Inverse Eigenvalue Problem ◽  
Nonnegative Matrices ◽  
Real Numbers ◽  
Inverse Eigenvalue ◽  
The Given

AbstractIn this paper, at first for a given set of real numbers with only one positive number, and in continue for a given set of real numbers in special conditions, we construct a symmetric nonnegative matrix such that the given set is its spectrum.

scholarly journals Explicit solution for an infinite dimensional generalized inverse eigenvalue problem

2001 ◽  
Vol 26 (9) ◽  
pp. 513-523 ◽  
Author(s):  
Kazem Ghanbari
Keyword(s):  
Eigenvalue Problem ◽  
Spectral Function ◽  
Spectral Data ◽  
Explicit Solution ◽  
Generalized Inverse ◽  
Jacobi Matrix ◽  
Inverse Eigenvalue Problem ◽  
Infinite Dimensional ◽  
Generalized Inverse Eigenvalue Problem ◽  
Inverse Eigenvalue

We study a generalized inverse eigenvalue problem (GIEP),Ax=λBx, in whichAis a semi-infinite Jacobi matrix with positive off-diagonal entriesci>0, andB= diag (b0,b1,…), wherebi≠0fori=0,1,…. We give an explicit solution by establishing an appropriate spectral function with respect to a given set of spectral data.

scholarly journals The Uniqueness of Solution to the Inverse Eigenvalue Problem for an Arrow-shaped Generalised Jacobi Matrix

2021 ◽  
Vol 2068 (1) ◽  
pp. 012014
Author(s):  
Hongliang Huang ◽  
Qike Wang ◽  
Zhibin Li ◽  
Lidong Wang
Keyword(s):  
Eigenvalue Problem ◽  
Uniqueness Theorem ◽  
Existence And Uniqueness ◽  
Jacobi Matrix ◽  
Inverse Eigenvalue Problem ◽  
Mathematical Expressions ◽  
Uniqueness Of The Solution ◽  
Inverse Eigenvalue ◽  
Mathematical Derivation ◽  
The Uniqueness Theorem

Abstract This paper studies the inverse eigenvalue problem for an arrow-shaped generalised Jacobi matrix, inverting matrices through two eigen-pairs. In the paper, the existence and uniqueness of the solution to the problem are discussed, and mathematical expressions as well as a numerical example are given. Finally, the uniqueness theorem of its matrix is established by mathematical derivation.

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.

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