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Least-squares solutions of matrix inverse problem for bi-symmetric matrices with a submatrix constraint. (English) Zbl 1199.65141

Summary: An \(n\times n\) real matrix \(A = (a_{ij})_{n\times n}\) is called bi-symmetric matrix if \(A\) is both symmetric and per-symmetric, that is, \(a_{ij} = a_{ji}\) and \(a_{ij} = a_{n+1-j,n+1-i}\) \((i,j = 1, 2,\dots , n)\). This paper is mainly concerned with finding the least-squares bi-symmetric solutions of matrix inverse problem \(AX = B\) with a submatrix constraint, where \(X\) and \(B\) are given matrices of suitable sizes. Moreover, in the corresponding solution set, the analytical expression of the optimal approximation solution to a given matrix \(A^*\) is derived. A direct method for finding the optimal approximation solution is described in detail, and three numerical examples are provided to show the validity of our algorithm.

MSC:

65F30 Other matrix algorithms (MSC2010)
15A24 Matrix equations and identities
65F20 Numerical solutions to overdetermined systems, pseudoinverses
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References:

[1] Linear Multivariable Control: A Geometric Approach. Springer: Berlin, 1979. · doi:10.1007/978-1-4684-0068-7
[2] Barcilon, Inverse Problems 3 pp 181– (1987)
[3] Inverse Problems in Vibration. Martinus Nijhoff: Dordrecht, The Netherlands, Boston, MA, 1986. · doi:10.1007/978-94-015-1178-0
[4] Joseph, AIAA Journal 30 pp 2890– (1992)
[5] Friedland, Journal of Mathematical Analysis and Applications 71 pp 412– (1979)
[6] Jiang, Mathematica Numerica Sinica 8 pp 47– (1986)
[7] Sun, Mathematica Numerica Sinica 3 pp 282– (1988)
[8] . Inverse Problem for Algebraic Eigenvalue. Henan Science and Technology Press: Zhengzhou, 1991.
[9] Li, Journal of China University of Science and Technology 14 pp 195– (1984)
[10] Liao, Numerical Mathematics–A Journal of Chinese University 7 pp 195– (1998)
[11] Mitra, Linear Algebra and Its Applications 131 pp 107– (1990)
[12] Wu, Linear Algebra and Its Applications 174 pp 145– (1992)
[13] Wu, Linear Algebra and Its Applications 236 pp 137– (1996)
[14] Zhang, Mathematica Numerica Sinica 11 pp 337– (1989)
[15] Woodgate, Linear Algebra and Its Applications 245 pp 171– (1996)
[16] Xie, Journal of Engineering Mathematics 4 pp 25– (1993)
[17] Zhou, Computers and Mathematics with Applications 45 pp 1581– (2003)
[18] , . Fast Algorithms of TOEPLITZ Form. Northwest Industry University Press: Xian, 1999 (in Chinese).
[19] Hu, Mathematica Numerica Sinica 4 pp 409– (1998)
[20] Liao, Mathematica Numerica Sinica 2 pp 209– (2001)
[21] Xie, Mathematica Numerica Sinica 1 pp 29– (2000)
[22] Xie, Journal of Computational Mathematics 6 pp 597– (2000)
[23] Peng, Numerical Linear Algebra with Applications 11 pp 59– (2004)
[24] Xu, Linear Algebra and Its Applications 279 pp 93– (1998)
[25] Golub, Linear Algebra and Its Applications 210 pp 3– (1994)
[26] Paige, SIAM Journal on Numerical Analysis 18 pp 398– (1981)
[27] Stewart, Numerische Mathematik 40 pp 297– (1982)
[28] . Matrix Perturbation Theory. Academic Press: New York, 1990. · Zbl 0706.65013
[29] Functional Analysis and Optimization Theory. Beijing University of Aeronautics and Astronautics Press: Beijing, 2003 (in chinese)
[30] Chang, Linear Algebra and Its Applications 179 pp 171– (1993)
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