The Hermitian \(R\)-conjugate generalized Procrustes problem. (English) Zbl 1291.15035

Summary: We consider the Hermitian \(R\)-conjugate generalized Procrustes problem to find Hermitian \(R\)-conjugate matrix \(X\) such that \(\sum^p_{k=1}||A_kX-C_k||^2+\sum^q_{l=1}||XB_l-D_l||^2\) is minimum, where \(A_k\), \(C_k\), \(B_l\), and \(D_l(k=1,2,\dots,p,l=1,\dots,q)\) are given complex matrices, and \(p\) and \(q\) are positive integers. The expression of the solution to Hermitian \(R\)-conjugate generalized Procrustes problem is derived. And the optimal approximation solution in the solution set for Hermitian \(R\)-conjugate generalized Procrustes problem to a given matrix is also obtained. Furthermore, we establish necessary and sufficient conditions for the existence and the formula for Hermitian \(R\)-conjugate solution to the linear system of complex matrix equations \(A_1X=C_1\), \(A_2X=C_2,\dots,A_pX=C_p\), \(XB_1=D_1,\dots,XB_q=D_q\) (\(p\) and \(q\) are positive integers). The representation of the corresponding optimal approximation problem is presented. Finally, an algorithm for solving two problems above is proposed, and the numerical examples show its feasibility.


15A24 Matrix equations and identities
65F30 Other matrix algorithms (MSC2010)
65K10 Numerical optimization and variational techniques
90C20 Quadratic programming
Full Text: DOI


[1] Green, B. F., The orthogonal approximation of an oblique structure in factor analysis, Psychometrika, 17, 429-440 (1952) · Zbl 0049.37601
[2] Higham, N. J., The symmetric Procrustes problem, BIT Numerical Mathematics, 28, 1, 133-143 (1988) · Zbl 0641.65034
[3] Peng, J.; Hu, X.-Y.; Zhang, L., The \((M, N)\)-symmetric Procrustes problem, Applied Mathematics and Computation, 198, 1, 24-34 (2008) · Zbl 1151.65034
[4] Trench, W. F., Hermitian, Hermitian \(R\)-symmetric, and Hermitian \(R\)-skew symmetric Procrustes problems, Linear Algebra and Its Applications, 387, 83-98 (2004) · Zbl 1121.15016
[5] Andersson, L.-E.; Elfving, T., A constrained Procrustes problem, SIAM Journal on Matrix Analysis and Applications, 18, 1, 124-139 (1997) · Zbl 0880.65017
[6] Gower, J. C., Generalized Procrustes analysis, Psychometrika, 40, 33-51 (1975) · Zbl 0305.62038
[7] Dai, H., On the symmetric solutions of linear matrix equations, Linear Algebra and Its Applications, 131, 1-7 (1990) · Zbl 0712.15009
[8] Peng, Z.-Y.; Hu, X.-Y., The reflexive and anti-reflexive solutions of the matrix equation \(A X = B\), Linear Algebra and Its Applications, 375, 147-155 (2003) · Zbl 1050.15016
[9] Horn, R. A.; Sergeichuk, V. V.; Shaked-Monderer, N., Solution of linear matrix equations in a \(^∗\) congruence class, Electronic Journal of Linear Algebra, 13, 153-156 (2005) · Zbl 1092.15010
[10] Hua, G. K.; Hu, X.; Zhang, L., A new iterative method for the matrix equation \(A X = B\), Applied Mathematics and Computation, 15, 1434-1441 (2007) · Zbl 1121.65043
[11] Porter, A. D., Solvability of the matrix equation \(A X = B\), Linear Algebra and Its Applications, 13, 3, 177-184 (1976) · Zbl 0333.15008
[12] Wang, Q.-W.; Yu, J., On the generalized bi (skew-) symmetric solutions of a linear matrix equation and its procrust problems, Applied Mathematics and Computation, 219, 19, 9872-9884 (2013) · Zbl 1290.15009
[13] Hanna, Y. S., On the solutions of tridiagonal linear systems, Applied Mathematics and Computation, 189, 2, 2011-2016 (2007) · Zbl 1122.65315
[14] Dong, C.-Z.; Wang, Q.-W.; Zhang, Y.-P., On the Hermitian \(R\)-conjugate solution of a system of matrix equations, Journal of Applied Mathematics, 2012 (2012) · Zbl 1268.15008
[15] Trench, W. F., Characterization and properties of matrices with generalized symmetry or skew symmetry, Linear Algebra and Its Applications, 377, 207-218 (2004) · Zbl 1046.15028
[16] Huckle, T.; Serra-Capizzano, S.; Tablino-Possio, C., Preconditioning strategies for Hermitian indefinite Toeplitz linear systems, SIAM Journal on Scientific Computing, 25, 5, 1633-1654 (2004) · Zbl 1067.65046
[17] Chan, R. H.; Yip, A. M.; Ng, M. K., The best circulant preconditioners for Hermitian Toeplitz systems, SIAM Journal on Numerical Analysis, 38, 3, 876-896 (2000) · Zbl 0978.65035
[18] Lee, A., Centro-Hermitian and skew-centro-Hermitian matrices, Linear Algebra and Its Applications, 29, 205-210 (1980) · Zbl 0435.15019
[19] Oppenheim, A. V., Applications of Digital Signal Processing (1978), Englewood Cliffs, Calif, USA: Prentice-Hall, Englewood Cliffs, Calif, USA
[20] Ng, M. K.; Plemmons, R. J.; Pimentel, F., A new approach to constrained total least squares image restoration, Linear Algebra and Its Applications, 316, 1-3, 237-258 (2000) · Zbl 0960.65044
[21] Kouassi, R.; Gouton, P.; Paindavoine, M., Approximation of the Karhunen-Loève tranformation and its application to colour images, Signal Processing: Image Commu, 16, 541-551 (2001)
[22] Deng, Y.-B.; Hu, X.-Y.; Zhang, L., Least squares solution of \(B X A^T = T\) over symmetric, skew-symmetric, and positive semidefinite \(X\), SIAM Journal on Matrix Analysis and Applications, 25, 2, 486-494 (2003) · Zbl 1050.65037
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