Left and right inverse eigenvalue problem of (R,S)-symmetric matrices and its optimal approximation problem

2013 ◽  
Vol 219 (17) ◽  
pp. 9261-9269 ◽  
Author(s):  
Feng Yin ◽  
Guang-Xin Huang
Keyword(s):  
Eigenvalue Problem ◽  
Approximation Problem ◽  
Inverse Eigenvalue Problem ◽  
Optimal Approximation ◽  
Symmetric Matrices ◽  
Inverse Eigenvalue ◽  
Left And Right

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 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.

The Inverse Eigenvalue Problem for Symmetric Doubly Arrow Matrices

2013 ◽  
Vol 444-445 ◽  
pp. 625-627
Author(s):  
Kan Ming Wang ◽  
Zhi Bing Liu ◽  
Xu Yun Fei
Keyword(s):  
Eigenvalue Problem ◽  
Special Kind ◽  
Inverse Eigenvalue Problem ◽  
Symmetric Matrices ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Inverse Eigenvalue ◽  
Necessary And Sufficient

In this paper we present a special kind of real symmetric matrices: the real symmetric doubly arrow matrices. That is, matrices which look like two arrow matrices, forward and backward, with heads against each other at the station, . We study a kind of inverse eigenvalue problem and give a necessary and sufficient condition for the existence of such matrices.

scholarly journals Achievable multiplicity partitions in the inverse eigenvalue problem of a graph

2019 ◽  
Vol 7 (1) ◽  
pp. 276-290
Author(s):  
Mohammad Adm ◽  
Shaun Fallat ◽  
Karen Meagher ◽  
Shahla Nasserasr ◽  
Sarah Plosker ◽  
...  
Keyword(s):  
Eigenvalue Problem ◽  
Inverse Eigenvalue Problem ◽  
Symmetric Matrices ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
Inverse Eigenvalue ◽  
Complete Multipartite

Abstract Associated to a graph G is a set 𝒮(G) of all real-valued symmetric matrices whose off-diagonal entries are nonzero precisely when the corresponding vertices of the graph are adjacent, and the diagonal entries are free to be chosen. If G has n vertices, then the multiplicities of the eigenvalues of any matrix in 𝒮 (G) partition n; this is called a multiplicity partition. We study graphs for which a multiplicity partition with only two integers is possible. The graphs G for which there is a matrix in 𝒮 (G) with partitions [n − 2, 2] have been characterized. We find families of graphs G for which there is a matrix in 𝒮 (G) with multiplicity partition [n − k, k] for k ≥ 2. We focus on generalizations of the complete multipartite graphs. We provide some methods to construct families of graphs with given multiplicity partitions starting from smaller such graphs. We also give constructions for graphs with matrix in 𝒮 (G) with multiplicity partition [n − k, k] to show the complexities of characterizing these graphs.

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