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.

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,

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.

SUFFICIENT CONDITIONS FOR THE SOLUBILITY OF ADDITIVE INVERSE EIGENVALUE PROBLEMS

1 IntroductionLet Rn×n be the set of all n×n real matrices.Rn=Rn×1.Cn×ndenotes the set of all n×n complex matrices.We are interested in solving the following inverse eigenvalue prob-lems:Problem A （Additive inverse eigenvalue problem） Given an n×n real matrix A=(aij),and n distinct real numbers λ1,λ2,…,λn,find a real n×n diagonal matrix

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.

Generalized Inverse Eigenvalue Problem for (P,Q)-Conjugate Matrices and the Associated Approximation Problem

In this paper,the generalized inverse eigenvalue problem for the（P,Q）-conjugate matrices and the associated approximation problem are discussed by using generalized singular value decomposition（GSVD）.Moreover,the least residual problem of the above generalized inverse eigenvalue problem is studied by using the canonical correlation decomposition（CCD）.The solutions to these problems are derived.Some numerical examples are given to illustrate the main results.

A SMALLEST SINGULAR VALUE METHOD FOR SOLVING INVERSE EIGENVALUE PROBLEMS

Utilizing the properties of the smallest singular value of a matrix, we propose a new, efficient and reliable algorithm for solving nonsymmetric matrix inverse eigenvalue problems, and compare it with a known method. We also present numerical experiments which illustrate our results.

A NOTE ON THE BACKWARD ERRORS FOR INVERSE EIGENVALUE PROBLEMS

In this note, we consider the backward errors for more general inverse eigenvalue problems by extending Sun's approach. The optimal backward errors are defined for diagonal-ization matrix inverse eigenvalue problem with respect to an approximate solution, and the upper and lower bounds are derived for the optimal backward errors. The results may be useful for testing the stability of practical algorithms.

Inverse Eigenvalue Problems for Exploring the Dynamics of Systems Biology Models

This paper describes inverse eigenvalue problems that arise in studying qualitative dynamics in systems biology models.An algorithm based on lift-andproject iterations is proposed,where the lifting step entails solving a constrained matrix inverse eigenvalue problem.In particular,prior to carrying out the iterative steps,a-priori bounds on the entries of the Jacobian matrix are computed by relying on the reaction network structure as well as the form of the rate law expressions for the model under consideration.Numerical results on a number of models show that the proposed algorithm can be used to computationally explore the possible dynamical scenarios while identifying the important mechanisms via the use of sparsity-promoting regularization.

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.

PARAMETER IDENTIFICATION OF DYNAMIC MODELS USING A BAYES APPROACH

The Bayesian method of statistical analysis has been applied to the parameter identification problem. A method is presented to identify parameters of dynamic models with the Bayes estimators of measurement frequencies. This is based on the solution of an inverse generalized evaluate problem. The stochastic nature of test data is considered and a normal distribution is used for the measurement frequencies. An additional feature is that the engineer's confidence in the measurement frequencies is quantified and incorporated into the identification procedure. A numerical example demonstrates the efficiency of the method.

ACG-TypeMethod for InverseQuadratic Eigenvalue Problems in Model Updating of Structural Dynamics

In this paper we first present a CG-type method for inverse eigenvalue problem of constructing real and symmetric matrices M,D and K for the quadratic pencil Q(λ)=λ^(2)M+λD+K,so that Q(λ)has a prescribed subset of eigenvalues and eigenvectors.This method can determine the solvability of the inverse eigenvalue problem automatically.We then consider the least squares model for updating a quadratic pencil Q(λ).More precisely,we update the model coefficient matrices M,C and K so that(i)the updated model reproduces the measured data,(ii)the symmetry of the original model is preserved,and(iii)the difference between the analytical triplet(M,D,K)and the updated triplet(M_(new),D_(new),K_(new))is minimized.In this paper a computationally efficient method is provided for such model updating and numerical examples are given to illustrate the effectiveness of the proposed method.

ON THE LEAST SQUARES PROBLEM OF A MATRIXEQUATION

Least squares solution of F=PG with respect to positive semidefinite symmetric P is considered,a new necessary and sufficient condition for solvablity is given,and the expression of solution is derived in the some special cases. Based on the expression, the least spuares solution of an inverse eigenvalue problem for positive semidefinite symmetric matrices is also given.