Volume 6, Issue 1
Submatrix Constrained Inverse Eigenvalue Problem involving Generalised Centrohermitian Matrices in Vibrating Structural Model Correction

East Asian J. Appl. Math., 6 (2016), pp. 42-59.

Published online: 2018-02

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• Abstract

Generalised centrohermitian and skew-centrohermitian matrices arise in a variety of applications in different fields. Based on the vibrating structure equation $M$$\ddot{x}+(D+G)$$\dot{x}$+$Kx$=$f(t)$ 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.

• Keywords

Left and right inverse eigenvalue problem, optimal approximation problem, generalised centrohermitian matrix, submatrix constraint.

• AMS Subject Headings

65F18, 15A51, 15A18, 15A12

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@Article{EAJAM-6-42, author = {}, title = {Submatrix Constrained Inverse Eigenvalue Problem involving Generalised Centrohermitian Matrices in Vibrating Structural Model Correction}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {6}, number = {1}, pages = {42--59}, abstract = {

Generalised centrohermitian and skew-centrohermitian matrices arise in a variety of applications in different fields. Based on the vibrating structure equation $M$$\ddot{x}+(D+G)$$\dot{x}$+$Kx$=$f(t)$ 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.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.200715.181115a}, url = {http://global-sci.org/intro/article_detail/eajam/10772.html} }
TY - JOUR T1 - Submatrix Constrained Inverse Eigenvalue Problem involving Generalised Centrohermitian Matrices in Vibrating Structural Model Correction JO - East Asian Journal on Applied Mathematics VL - 1 SP - 42 EP - 59 PY - 2018 DA - 2018/02 SN - 6 DO - http://doi.org/10.4208/eajam.200715.181115a UR - https://global-sci.org/intro/article_detail/eajam/10772.html KW - Left and right inverse eigenvalue problem, optimal approximation problem, generalised centrohermitian matrix, submatrix constraint. AB -

Generalised centrohermitian and skew-centrohermitian matrices arise in a variety of applications in different fields. Based on the vibrating structure equation $M$$\ddot{x}+(D+G)$$\dot{x}$+$Kx$=$f(t)$ 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.

Wei-Ru Xu & Guo-Liang Chen. (2020). Submatrix Constrained Inverse Eigenvalue Problem involving Generalised Centrohermitian Matrices in Vibrating Structural Model Correction. East Asian Journal on Applied Mathematics. 6 (1). 42-59. doi:10.4208/eajam.200715.181115a
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