ABSTRACT

The structured partial quadratic inverse eigenvalue problem (SPQIEP) is to construct the structured quadratic matrix polynomial using the partial eigendata. The structures arising in physical applications include symmetry, band (tridiagonal, diagonal, pentagonal) etc. The construction of the structured matrix polynomial is the most difficult aspect of this problem and the research on structured inverse eigenvalue problem is rare. In this paper, the symmetric band partial quadratic inverse eigenvalue problem (SBPQIEP) for the damped mass spring system is considered. This problem concerns in finding the symmetric band matrices $M\in {\mathbb{R}}^{n\times n}$, $K\in {\mathbb{R}}^{n\times n}$ and C $\in {\mathbb{R}}^{n\times n}$ with bandwidth p from m ( $1\le m\le 2n$) prescribed eigenpairs so that the corresponding quadratic matrix polynomial $P(\lambda )={\lambda }^{2}M+\lambda C+K$ has the given eigenpairs as its eigenvalues and eigenvectors. In general, SBPQIEP is very hard to solve due to the additional band structure constraint. We propose a novel method, based on the matrix-vectorization and Kronecker product of matrices for solving this problem. Furthermore, explicit expressions for general solutions are presented. Numerical experiments on a spring mass problem are presented to illustrate the applicability and the practical usefulness of the proposed method.

摘要

结构化部分二次特征值逆问题 (SPQIEP) 是利用部分特征数据构造结构化二次矩阵多项式。物理应用中出现的结构包括对称性、带状（三对角线、对角线、五边形）等。结构化矩阵多项式的构造是该问题最困难的方面，对结构化特征值反问题的研究很少。在本文中，考虑了阻尼质量弹簧系统的对称带部分二次特征值反问题（SBPQIEP）。这个问题涉及寻找对称带矩阵$米\in {\mathbb{电阻}}^{n\times n}$, $钾\in {\mathbb{电阻}}^{n\times n}$和C $\in {\mathbb{电阻}}^{n\times n}$带宽p来自m ($1\le 米\le 2n$) 规定的特征对使得相应的二次矩阵多项式 $磷(\lambda )={\lambda }^2米+\lambda C+钾$具有给定的特征对作为其特征值和特征向量。通常，由于附加的能带结构约束，SBPQIEP 很难求解。我们提出了一种基于矩阵向量化和 Kronecker 乘积的新方法来解决这个问题。此外，还给出了一般解决方案的显式表达式。提出了关于弹簧质量问题的数值实验，以说明所提出方法的适用性和实用性。