Least squares solutions to \(AX = B\) for bisymmetric matrices under a central principal submatrix constraint and the optimal approximation. (English) Zbl 1133.15016

A square matrix is bisymmetric if it is symmetric w.r.t. the diagonal and the antidiagonal. The authors consider least squares solutions to the matrix equation \(AX=B\) under a central principal matrix constaint for \(A\) and the optimal approximation. A central principal submatrix is obtained by deleting the same number of rows and columns in the edges. The authors first discuss the structure of bisymmetric matrices and their principal submatrices. Then they give necessary and sufficient conditions for the solvability of the least squares problem, and derive the general representation of the solutions. They express also the solution to the corresponding optimal approximation problem.


15A24 Matrix equations and identities
15B57 Hermitian, skew-Hermitian, and related matrices
65F20 Numerical solutions to overdetermined systems, pseudoinverses
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[1] Deift, P.; Nanda, T., On the determination of a tridiagonal matrix from its spectrum and a submatrix, Linear Algebra Appl., 60, 43-55 (1984) · Zbl 0541.15004
[2] Peng, Z. Y.; Hu, X. Y., Constructing Jacobi matrix with prescribed ordered defective eigenpairs and a principal submatrix, J. Comput. Appl. Math., 175, 321-333 (2005) · Zbl 1066.65043
[3] Peng, Z. Y.; Hu, X. Y.; Zhang, L., The inverse problem of bisymmetric matrices with a submatrix constraint, Numer. Linear Algebra Appl., 11, 59-73 (2004) · Zbl 1164.15322
[4] Gong, L. S.; Hu, X. Y.; Zhang, L., The expansion problem of anti-symmetric matrix under a linear constraint and the optimal approximation, J. Comput. Appl. Math., 197, 44-52 (2006) · Zbl 1103.15008
[5] Sun, J. G., Two kinds of inverse eigenvalue problems for real symmetric matrices, Math. Numer. Sin., 10, 3, 282-290 (1988) · Zbl 0656.65041
[6] Xie, D. X.; Zhang, L.; Hu, X. Y., The solvability conditions for the inverse problem of bisymmetric nonnegative definite matrices, J. Comput. Math., 18, 6, 597-608 (2000) · Zbl 0966.15008
[7] Wang, Q. W., Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations, Comput. Math. Appl., 49, 641-650 (2005) · Zbl 1138.15003
[8] Yin, Q. X., Construction of real antisymmetric and bi-antisymmetric matrices with prescribed spectrum data, Linear Algebra Appl., 389, 95-106 (2004) · Zbl 1069.15013
[9] Dai, H.; Peter, L., Linear matrix equations from an inverse problem of vibration theory, Linear Algebra Appl., 246, 31-47 (1996) · Zbl 0861.15014
[10] Cantoni, A.; Butler, P., Eigenvalues and eigenvectors of symmetric centrosymmetric matrices, Linear Algebra Appl., 13, 275-288 (1976) · Zbl 0326.15007
[11] Cantoni, A.; Butler, P., Properties of the eigenvectors of persymmetric matrices with applications to communication theory, IEEE Trans. Commun. COM, 24, 8, 804-809 (1976) · Zbl 0351.94001
[12] Melman, A., Symmetric centrosymmetric matrix-vector multiplication, Linear Algebra Appl., 320, 193-198 (2000) · Zbl 0971.65022
[13] O. Hald, On discrete and numerical Sturm-Liouville problems, Ph.D. dissertation, Department of Mathematics, New York University, New York, 1972.; O. Hald, On discrete and numerical Sturm-Liouville problems, Ph.D. dissertation, Department of Mathematics, New York University, New York, 1972.
[14] Fausett, D. W.; Fulton, C. T., Large least squares problems involving Kronecker products, SIAM J. Matrix Anal. Appl., 15, 1, 219-227 (1994) · Zbl 0798.65059
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