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Volume 18, Issue 6
The Solvability Conditions for the Inverse Problem of Bisymmetric Nonnegative Definite Matrices

Dong-Xiu Xie, Lei Zhang & Xi-Yan Hu

J. Comp. Math., 18 (2000), pp. 597-608.

Published online: 2000-12

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  • Abstract

$A = (a_{ij}) \in R^{n×n}$ is termed bisymmetric matrix if $$a_{ij} = a_{ji} = a_{n-j+1,n-i+1}, i,j=1,2, ..., n.$$ We denote the set of all $n \times n$ bisymmetric matrices by $BSR^{n×n}$.
This paper is mainly concerned with solving the following two problems:
Problem I. Given $X, B \in R^{n×m}$, find $A \in P_n$ such that $AX=B$, where $P_n = \{ A \in BSR^{n×n}| x^TAx \ge 0, \forall x \in R^n \}$.
Problem Ⅱ. Given $A^* \in R^{n×n}$, find $\hat{A} \in S_E$ such that $$\| A^* -\hat{A}\|_F =\mathop{min}\limits_{A \in S_E} \| A^* - A \|_F,$$ where $\|\cdot\|_F$ is Frobenius norm, and $S_E$ denotes the solution set of problem I.

The necessary and sufficient conditions for the solvability of problem I have been studied. The general form of $S_E$ has been given. For problem II the expression of the solution has been provided.

  • Keywords

Frobenius norm, Bisymmetric matrix, The optimal solution.

  • AMS Subject Headings

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COPYRIGHT: © Global Science Press

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@Article{JCM-18-597, author = {Xie , Dong-XiuZhang , Lei and Hu , Xi-Yan}, title = {The Solvability Conditions for the Inverse Problem of Bisymmetric Nonnegative Definite Matrices}, journal = {Journal of Computational Mathematics}, year = {2000}, volume = {18}, number = {6}, pages = {597--608}, abstract = {

$A = (a_{ij}) \in R^{n×n}$ is termed bisymmetric matrix if $$a_{ij} = a_{ji} = a_{n-j+1,n-i+1}, i,j=1,2, ..., n.$$ We denote the set of all $n \times n$ bisymmetric matrices by $BSR^{n×n}$.
This paper is mainly concerned with solving the following two problems:
Problem I. Given $X, B \in R^{n×m}$, find $A \in P_n$ such that $AX=B$, where $P_n = \{ A \in BSR^{n×n}| x^TAx \ge 0, \forall x \in R^n \}$.
Problem Ⅱ. Given $A^* \in R^{n×n}$, find $\hat{A} \in S_E$ such that $$\| A^* -\hat{A}\|_F =\mathop{min}\limits_{A \in S_E} \| A^* - A \|_F,$$ where $\|\cdot\|_F$ is Frobenius norm, and $S_E$ denotes the solution set of problem I.

The necessary and sufficient conditions for the solvability of problem I have been studied. The general form of $S_E$ has been given. For problem II the expression of the solution has been provided.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9070.html} }
TY - JOUR T1 - The Solvability Conditions for the Inverse Problem of Bisymmetric Nonnegative Definite Matrices AU - Xie , Dong-Xiu AU - Zhang , Lei AU - Hu , Xi-Yan JO - Journal of Computational Mathematics VL - 6 SP - 597 EP - 608 PY - 2000 DA - 2000/12 SN - 18 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9070.html KW - Frobenius norm, Bisymmetric matrix, The optimal solution. AB -

$A = (a_{ij}) \in R^{n×n}$ is termed bisymmetric matrix if $$a_{ij} = a_{ji} = a_{n-j+1,n-i+1}, i,j=1,2, ..., n.$$ We denote the set of all $n \times n$ bisymmetric matrices by $BSR^{n×n}$.
This paper is mainly concerned with solving the following two problems:
Problem I. Given $X, B \in R^{n×m}$, find $A \in P_n$ such that $AX=B$, where $P_n = \{ A \in BSR^{n×n}| x^TAx \ge 0, \forall x \in R^n \}$.
Problem Ⅱ. Given $A^* \in R^{n×n}$, find $\hat{A} \in S_E$ such that $$\| A^* -\hat{A}\|_F =\mathop{min}\limits_{A \in S_E} \| A^* - A \|_F,$$ where $\|\cdot\|_F$ is Frobenius norm, and $S_E$ denotes the solution set of problem I.

The necessary and sufficient conditions for the solvability of problem I have been studied. The general form of $S_E$ has been given. For problem II the expression of the solution has been provided.

Dong-Xiu Xie, Lei Zhang & Xi-Yan Hu. (1970). The Solvability Conditions for the Inverse Problem of Bisymmetric Nonnegative Definite Matrices. Journal of Computational Mathematics. 18 (6). 597-608. doi:
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