A STABILITY ANALYSIS OF THE (k) JACOBI MATRIX INVERSE EIGENVALUE PROBLEM

In this paper we will analyze the perturbation quality for a new algorithm of the (k) Jacobi matrix inverse eigenvalue problem.

A new algorithm for an inverse eigenvalue problem on Jacobi matrices

In this paper, an inverse problem on Jacobi matrices presented by Shieh in 2004 is studied. Shieh＇s result is improved and a new and stable algorithm to reconstruct its solution is given. The numerical examples is also given.

AN INVERSE EIGENVALUE PROBLEM FOR JACOBI MATRICES

is gained by deleting the kth row and the kth column （k=1,2,...,n） from Tn.We put for-ward an inverse eigenvalue problem to be that:If we don’t know the matrix T1,n,but weknow all eigenvalues of matrix T1,k-1,all eigenvalues of matrix Tk+1,k,and all eigenvaluesof matrix T1,n could we construct the matrix T1,n.Let μ1,μ2,…,μk-1,μk,μk+1,…,μn-1,

THE UNSOLVABILITY OF GENERALIZED INVERSE EIGENVALUE PROBLEMS ALMOST EVERYWHERE

In this paper the unsolvability of generalized inverse eigenvalue problems almost everywhere is discussed.We first give the definitions for the unsolvability of generalized inverse eigenvalue problems almost everywhere.Then adopting the method used in [14],we present some sufficient conditions such that the generalized inverse eigenvalue problems are unsohable almost everywhere.

THE BI-SELF-CONJUGATE AND NONNEGATIVE DEFINITE SOLUTIONS TO THE INVERSE EIGENVALUE PROBLEM OF QUATERNION MATRICES

The main aim of this paper is to discuss the following two problems: Problem I: Given X ∈ Hn×m (the set of all n×m quaternion matrices), A = diag(λ1,…, λm) EEEEE Hm×m, find A ∈ BSHn×n≥such that AX = X(?), where BSHn×n≥ denotes the set of all n×n quaternion matrices which are bi-self-conjugate and nonnegative definite. Problem Ⅱ2= Given B ∈ Hn×m, find B ∈ SE such that ||B-B||Q = minAE∈=sE ||B-A||Q, where SE is the solution set of problem I , || ·||Q is the quaternion matrix norm. The necessary and sufficient conditions for SE being nonempty are obtained. The general form of elements in SE and the expression of the unique solution B of problem Ⅱ are given.

Generalized Inverse Eigenvalue Problem for Centrohermitian Matrices

In this paper we first consider the existence and the general form of solution to the following generalized inverse eigenvalue problem(GIEP): given a set of n-dimension complex vectors {x j}m j=1 and a set of complex numbers {λ j}m j=1, find two n×n centrohermitian matrices A,B such that {x j}m j=1 and {λ j}m j=1 are the generalized eigenvectors and generalized eigenvalues of Ax=λBx, respectively. We then discuss the optimal approximation problem for the GIEP. More concretely, given two arbitrary matrices, , ∈C n×n, we find two matrices A and B such that the matrix (A*,B*) is closest to (,) in the Frobenius norm, where the matrix (A*,B*) is the solution to the GIEP. We show that the expression of the solution of the optimal approximation is unique and derive the expression for it.

On Open Problems of Nonnegative Inverse Eigenvalues Problem

In this paper, we give solvability conditions for three open problems of nonnegative inverse eigenvalues problem (NIEP) which were left hanging in the air up to seventy years. It will offer effective ways to judge an NIEP whether is solvable.

Solvability conditions for algebra inverse eigenvalue problem over set of anti-Hermitian generalized anti-Hamiltonian matrices

By using the characteristic properties of the anti-Hermitian generalized anti-Hamiltonian matrices, we prove some necessary and sufficient conditions of the solvability for algebra inverse eigenvalue problem of anti-Hermitian generalized anti-Hamiltonian matrices, and obtain a general expression of the solution to this problem. By using the properties of the orthogonal projection matrix, we also obtain the expression of the solution to optimal approximate problem of an n× n complex matrix under spectral restriction.

A Solution of Inverse Eigenvalue Problems for Unitary Hessenberg Matrices

Let H∈Cn×n be an n×n unitary upper Hessenberg matrix whose subdiagonal elements are all positive. Partition H as H=[H11 H12 H21 H22],(0.1) where H11 is its k×k leading principal submatrix; H22 is the complementary matrix of H11. In this paper, H is constructed uniquely when its eigenvalues and the eigenvalues of (H|^)11 and (H|^)22 are known. Here (H|^)11 and (H|^)22 are rank-one modifications of H11 and H22 respectively.

Solubility Existence of Inverse Eigenvalue Problem for a Class of Singular Hermitian Matrices

In this article,we discuss singular Hermitian matrices of rank greater or equal to four for an inverse eigenvalue problem.Specifically,we look into how to generate n by n singular Hermitian matrices of ranks four and five from a prescribed spectrum.Numerical examples are presented in each case to illustrate these scenarios.It was established that given a prescribed spectral datum and it multiplies,then the solubility of the inverse eigenvalue problem for n by n singular Hermitian matrices of rank r exists.

一类周期伪Jacobi矩阵的逆特征值问题

该文考虑了一类周期伪Jacobi矩阵的逆特征值问题,该矩阵依赖于一个符号算子,该符号算子分量的变化将会对整个矩阵的谱造成很大的扰动.于是根据该矩阵特征方程根的分布情况来讨论其特征值的分布.当该符号算子中最后一个分量发生变化时,给出了其逆特征值问题可解的充要条件和具体的构造过程.最后,通过数值算例验证了所给算法的有效性和可行性.

Solution of an inverse problem for“fixed-fixed”and“fixed-free”spring-mass systems

The tridiagonal coefficient matrix for the ＂fixed-fixed＂ spring-mass system was obtained by changing spring length. And then a new algorithm of the inverse problem was designed to construct the masses and the spring constants from the natural frequencies of the ＂fixed-fixed＂ and ＂fixed-fres＂ spring-mass systems. An example was given to illustrate the results.

Eigenvalues of Jacobian Matrices Report on Steps of Metabolic Reprogramming in a Complex Plant-Environment Interaction

Mathematical modeling of biochemical systems aims at improving the knowledge about complex regulatory networks. The experimental high-throughput measurement of levels of biochemical components, like metabolites and proteins, has become an integral part for characterization of biological systems. Yet, strategies of mathematical modeling to functionally integrate resulting data sets is still challenging. In plant biology, regulatory strategies that determine the metabolic output of metabolism as a response to changes in environmental conditions are hardly traceable by intuition. Mathematical modeling has been shown to be a promising approach to address such problems of plant-environment interaction promoting the comprehensive understanding of plant biochemistry and physiology. In this context, we recently published an inversely calculated solution for first-order partial derivatives, i.e. the Jacobian matrix, from experimental high-throughput data of a plant biochemical model system. Here, we present a biomathematical strategy, comprising 1) the inverse calculation of a biochemical Jacobian;2) the characterization of the associated eigenvalues and 3) the interpretation of the results with respect to biochemical regulation. Deriving the real parts of eigenvalues provides information about the stability of solutions of inverse calculations. We found that shifts of the eigenvalue real part distributions occur together with metabolic shifts induced by short-term and long-term exposure to low temperature. This indicates the suitability of mathematical Jacobian characterization for recognizing perturbations in the metabolic homeostasis of plant metabolism. Together with our previously published results on inverse Jacobian calculation this represents a comprehensive strategy of mathematical modeling for the analysis of complex biochemical systems and plant-environment interactions from the molecular to the ecosystems level.

AN INVERSE EIGENVALUE PROBLEM FOR JACOBI MATRICES

Let T1,n be an n x n unreduced symmetric tridiagonal matrix with eigenvaluesand is an (n - 1) x (n - 1) submatrix by deleting the kth row and kth column, k = 1, 2,be the eigenvalues of T1,k andbe the eigenvalues of Tk+1,nA new inverse eigenvalues problem has put forward as follows: How do we construct anunreduced symmetric tridiagonal matrix T1,n, if we only know the spectral data: theeigenvalues of T1,n, the eigenvalues of Ti,k-1 and the eigenvalues of Tk+1,n?Namely if we only know the data: A1, A2, An,how do we find the matrix T1,n? A necessary and sufficient condition and an algorithm ofsolving such problem, are given in this paper.

AN INVERSE EIGENVALUE PROBLEM FOR JACOBI MATRICES

In this paper, we discuss an inverse eigenvalue problem for constructing a 2n × 2n Jacobi matrix T such that its 2n eigenvalues are given distinct real values and its leading principal submatrix of order n is a given Jacobi matrix. A new sufficient and necessary condition for the solvability of the above problem is given in this paper. Furthermore, we present a new algorithm and give some numerical results.

ON THE CONSTRUCTION OF A JACOBI MATRIX FROM ITS SPECTRUM AND A SUBMATRIX

Let J. be an n × n Jacobi matrix and Al λ1 ,..',λ2n distinct real numbers. The following problem is well known. Under what condition does there exist a 2n × 2n Jacobi matrix J. such that J,. has eigenvalues λ1, ..λ2n A. and its leading n × n principal submatrix is exactly Jn In this paper a condition for the solubility of the problem is given. The dependence of J2n on the given data is shown to be continuous.

子周期Jacobi矩阵特征值反问题

讨论了一类由谱数据构造子周期Jacobi矩阵的特征值反问题,以及子周期Jacobi矩阵的相关性质.得到了问题有解的充分必要条件以及有唯一解的充分必要条件,并且给出了可行稳定的数值算法.

一类广义Jacobi矩阵的逆特征值问题

本文研究了一类广义Jacobi矩阵的逆特征值问题,给出了该问题有解的充要条件,并讨论了解的唯一性.进一步,本文给出算法计算该问题的解,数值实例说明算法是行之有效的.

一类Jacobi矩阵特征值反问题

给定三个互异实数 α,β,γ及三个不同的非零实向量 x=( x1 ,x2 ,… ,xn) T,y=( y1 ,y2 ,… ,yn) T,z=( z1 ,z2 ,… ,zn) T,构造 n阶 Jacobi矩阵 J使 ( α,x) ,( β,y) ,( γ,z)是 J的第 p,q,r个特征对。给出了这一类

广义Jacobi矩阵特征值反问题

提出广义Jacobi矩阵特征值反问题,也就是次对角线元素乘积为正的Jacobi矩阵的特征值反问题,问题IEPGJM:①给定两个互异实数λ,μ(λ<μ)和两个n维非零实向量x,y,求n阶实广义Jacobi矩阵J=[ai,bi,kbi],使得Jx=λx,Jy=μy;②给定3个互异实数λ,μ,γ,和3个n维非零实向量x,y,z,求n阶实广义Jacobi矩阵J=[ai,bi,ci],使得Jx=λx,Jy=μy,Jz=γz.文中给出了问题解的表达式,提供了一个数值例子.