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Volume 22, Issue 6
The Nearest Bisymmetric Solutions of Linear Matrix Equations

Zhenyun Peng, Xiyan Hu & Lei Zhang

J. Comp. Math., 22 (2004), pp. 873-880.

Published online: 2004-12

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  • Abstract

The necessary and sufficient conditions for the existence of and the expressions for the bisymmetric solutions of the matrix equations (I) $A_1X_1B_1+A_2X_2B_2+\cdots+A_kX_kB_k=D$, (II) $A_1XB_1+A_2XB_2+\cdots+A_kXB_k=D$ and (III) $(A_1XB_1, A_2XB_2, ··· , A_kXB_k) = (D_1, D_2, ··· , D_k)$ are derived by using Kronecker product and Moore-Penrose generalized inverse of matrices. In addition, in corresponding solution set of the matrix equations, the explicit expression of the nearest matrix to a given matrix in the Frobenius norm is given. Numerical methods and numerical experiments of finding the nearest solutions are also provided.

  • Keywords

Bisymmetric matrix, Matrix equation, Matrix nearness problem, Kronecker product, Frobenius norm, Moore-Penrose generalized inverse.

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COPYRIGHT: © Global Science Press

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@Article{JCM-22-873, author = {Peng , ZhenyunHu , Xiyan and Zhang , Lei}, title = {The Nearest Bisymmetric Solutions of Linear Matrix Equations}, journal = {Journal of Computational Mathematics}, year = {2004}, volume = {22}, number = {6}, pages = {873--880}, abstract = {

The necessary and sufficient conditions for the existence of and the expressions for the bisymmetric solutions of the matrix equations (I) $A_1X_1B_1+A_2X_2B_2+\cdots+A_kX_kB_k=D$, (II) $A_1XB_1+A_2XB_2+\cdots+A_kXB_k=D$ and (III) $(A_1XB_1, A_2XB_2, ··· , A_kXB_k) = (D_1, D_2, ··· , D_k)$ are derived by using Kronecker product and Moore-Penrose generalized inverse of matrices. In addition, in corresponding solution set of the matrix equations, the explicit expression of the nearest matrix to a given matrix in the Frobenius norm is given. Numerical methods and numerical experiments of finding the nearest solutions are also provided.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/10291.html} }
TY - JOUR T1 - The Nearest Bisymmetric Solutions of Linear Matrix Equations AU - Peng , Zhenyun AU - Hu , Xiyan AU - Zhang , Lei JO - Journal of Computational Mathematics VL - 6 SP - 873 EP - 880 PY - 2004 DA - 2004/12 SN - 22 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/10291.html KW - Bisymmetric matrix, Matrix equation, Matrix nearness problem, Kronecker product, Frobenius norm, Moore-Penrose generalized inverse. AB -

The necessary and sufficient conditions for the existence of and the expressions for the bisymmetric solutions of the matrix equations (I) $A_1X_1B_1+A_2X_2B_2+\cdots+A_kX_kB_k=D$, (II) $A_1XB_1+A_2XB_2+\cdots+A_kXB_k=D$ and (III) $(A_1XB_1, A_2XB_2, ··· , A_kXB_k) = (D_1, D_2, ··· , D_k)$ are derived by using Kronecker product and Moore-Penrose generalized inverse of matrices. In addition, in corresponding solution set of the matrix equations, the explicit expression of the nearest matrix to a given matrix in the Frobenius norm is given. Numerical methods and numerical experiments of finding the nearest solutions are also provided.

Peng , ZhenyunHu , Xiyan and Zhang , Lei. (2004). The Nearest Bisymmetric Solutions of Linear Matrix Equations. Journal of Computational Mathematics. 22 (6). 873-880. doi:
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