An inverse eigenvalue problem and a matrix approximation problem for symmetric skew-hamiltonian matrices

2007 ◽  
Vol 46 (1) ◽  
pp. 23-34 ◽  
Author(s):  
Dongxiu Xie ◽  
Ningjun Huang ◽  
Qin Zhang
Keyword(s):  
Eigenvalue Problem ◽  
Approximation Problem ◽  
Inverse Eigenvalue Problem ◽  
Matrix Approximation ◽  
Inverse Eigenvalue

scholarly journals A class of constrained inverse eigenvalue problem and associated approximation problem for symmetrizable matrices

Filomat ◽  
2019 ◽  
Vol 33 (7) ◽  
pp. 1903-1909
Author(s):  
Xiangyang Peng ◽  
Wei Liu ◽  
Jinrong Shen
Keyword(s):  
Eigenvalue Problem ◽  
Symmetric Matrix ◽  
Approximation Problem ◽  
Inverse Eigenvalue Problem ◽  
Optimal Approximation ◽  
Original Matrix ◽  
Approximation Solution ◽  
Solvability Conditions ◽  
Inverse Eigenvalue ◽  
Symmetrizable Matrices

The real symmetric matrix is widely applied in various fields, transforming non-symmetric matrix to symmetric matrix becomes very important for solving the problems associated with the original matrix. In this paper, we consider the constrained inverse eigenvalue problem for symmetrizable matrices, and obtain the solvability conditions and the general expression of the solutions. Moreover, we consider the corresponding optimal approximation problem, obtain the explicit expressions of the optimal approximation solution and the minimum norm solution, and give the algorithm and corresponding computational example.

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 Left and right inverse eigenpairs problem with a submatrix constraint for the generalized centrosymmetric matrix

2020 ◽  
Vol 18 (1) ◽  
pp. 603-615
Author(s):  
Fan-Liang Li
Keyword(s):  
Eigenvalue Problem ◽  
Approximation Problem ◽  
Inverse Eigenvalue Problem ◽  
Optimal Approximation ◽  
Frobenius Norm ◽  
Approximation Solution ◽  
Inverse Eigenvalue ◽  
Left And Right ◽  
Centrosymmetric Matrix ◽  
Value Decomposition

Abstract Left and right inverse eigenpairs problem is a special inverse eigenvalue problem. There are many meaningful results about this problem. However, few authors have considered the left and right inverse eigenpairs problem with a submatrix constraint. In this article, we will consider the left and right inverse eigenpairs problem with the leading principal submatrix constraint for the generalized centrosymmetric matrix and its optimal approximation problem. Combining the special properties of left and right eigenpairs and the generalized singular value decomposition, we derive the solvability conditions of the problem and its general solutions. With the invariance of the Frobenius norm under orthogonal transformations, we obtain the unique solution of optimal approximation problem. We present an algorithm and numerical experiment to give the optimal approximation solution. Our results extend and unify many results for left and right inverse eigenpairs problem and the inverse eigenvalue problem of centrosymmetric matrices with a submatrix constraint.

scholarly journals An Inverse Eigenvalue Problem of Hermite-Hamilton Matrices in Structural Dynamic Model Updating

2010 ◽  
Vol 2010 ◽  
pp. 1-11
Author(s):  
Linlin Zhao ◽  
Guoliang Chen
Keyword(s):  
Eigenvalue Problem ◽  
Generalized Inverse ◽  
Model Updating ◽  
Approximation Problem ◽  
Inverse Eigenvalue Problem ◽  
Structural Dynamic ◽  
Solvability Conditions ◽  
Structural Dynamic Model ◽  
Singular Value Decompositions ◽  
Inverse Eigenvalue

We first consider the following inverse eigenvalue problem: givenX∈Cn×mand a diagonal matrixΛ∈Cm×m, findn×nHermite-Hamilton matricesKandMsuch thatKX=MXΛ. We then consider an optimal approximation problem: givenn×nHermitian matricesKaandMa, find a solution(K,M)of the above inverse problem such that∥K-Ka∥2+∥M-Ma∥2=min⁡. By using the Moore-Penrose generalized inverse and the singular value decompositions, the solvability conditions and the representations of the general solution for the first problem are derived. The expression of the solution to the second problem is presented.

scholarly journals An inverse eigenvalue problem for modified pseudo-Jacobi matrices

2021 ◽  
Vol 389 ◽  
pp. 113361
Author(s):  
Wei-Ru Xu ◽  
Natália Bebiano ◽  
Guo-Liang Chen
Keyword(s):  
Eigenvalue Problem ◽  
Inverse Eigenvalue Problem ◽  
Jacobi Matrices ◽  
Inverse Eigenvalue

scholarly journals Toeplitz nonnegative realization of spectra via companion matrices

2019 ◽  
Vol 7 (1) ◽  
pp. 230-245
Author(s):  
Macarena Collao ◽  
Mario Salas ◽  
Ricardo L. Soto
Keyword(s):  
Eigenvalue Problem ◽  
Half Plane ◽  
Toeplitz Matrix ◽  
Toeplitz Matrices ◽  
Nonnegative Matrix ◽  
Inverse Eigenvalue Problem ◽  
Complex Numbers ◽  
Inverse Eigenvalue ◽  
Companion Matrices ◽  
Solution Matrix

Abstract The nonnegative inverse eigenvalue problem (NIEP) is the problem of finding conditions for the existence of an n × n entrywise nonnegative matrix A with prescribed spectrum Λ = {λ1, . . ., λn}. If the problem has a solution, we say that Λ is realizable and that A is a realizing matrix. In this paper we consider the NIEP for a Toeplitz realizing matrix A, and as far as we know, this is the first work which addresses the Toeplitz nonnegative realization of spectra. We show that nonnegative companion matrices are similar to nonnegative Toeplitz ones. We note that, as a consequence, a realizable list Λ= {λ1, . . ., λn} of complex numbers in the left-half plane, that is, with Re λi≤ 0, i = 2, . . ., n, is in particular realizable by a Toeplitz matrix. Moreover, we show how to construct symmetric nonnegative block Toeplitz matrices with prescribed spectrum and we explore the universal realizability of lists, which are realizable by this kind of matrices. We also propose a Matlab Toeplitz routine to compute a Toeplitz solution matrix.

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