Abstract and Applied Analysis

The Hermitian R -Conjugate Generalized Procrustes Problem

Hai-Xia Chang, Xue-Feng Duan, and Qing-Wen Wang

Full-text: Open access

Abstract

We consider the Hermitian R -conjugate generalized Procrustes problem to find Hermitian R -conjugate matrix X such that k = 1 p A k X - C k 2  +  l = 1 q X B l - D l 2 is minimum, where A k , C k , B l , and D l ( k = 1,2 , , p , l = 1 , , 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 1 X = C 1 , A 2 X = C 2 , , A p X = C p , X B 1 = D 1 , , X B 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.

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 423605, 9 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393447686

Digital Object Identifier
doi:10.1155/2013/423605

Mathematical Reviews number (MathSciNet)
MR3108633

Zentralblatt MATH identifier
1291.15035

Citation

Chang, Hai-Xia; Duan, Xue-Feng; Wang, Qing-Wen. The Hermitian $R$ -Conjugate Generalized Procrustes Problem. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 423605, 9 pages. doi:10.1155/2013/423605. https://projecteuclid.org/euclid.aaa/1393447686


Export citation

References

  • B. F. Green, “The orthogonal approximation of an oblique structure in factor analysis,” Psychometrika, vol. 17, pp. 429–440, 1952.
  • N. J. Higham, “The symmetric Procrustes problem,” BIT Numerical Mathematics, vol. 28, no. 1, pp. 133–143, 1988.
  • J. Peng, X.-Y. Hu, and L. Zhang, “The $(M,N)$-symmetric Procrustes problem,” Applied Mathematics and Computation, vol. 198, no. 1, pp. 24–34, 2008.
  • W. F. Trench, “Hermitian, Hermitian \emphR-symmetric, and Hermitian \emphR-skew symmetric Procrustes problems,” Linear Algebra and Its Applications, vol. 387, pp. 83–98, 2004.
  • L.-E. Andersson and T. Elfving, “A constrained Procrustes problem,” SIAM Journal on Matrix Analysis and Applications, vol. 18, no. 1, pp. 124–139, 1997.
  • J. C. Gower, “Generalized Procrustes analysis,” Psychometrika, vol. 40, pp. 33–51, 1975.
  • H. Dai, “On the symmetric solutions of linear matrix equations,” Linear Algebra and Its Applications, vol. 131, pp. 1–7, 1990.
  • Z.-Y. Peng and X.-Y. Hu, “The reflexive and anti-reflexive solutions of the matrix equation $AX=B$,” Linear Algebra and Its Applications, vol. 375, pp. 147–155, 2003.
  • R. A. Horn, V. V. Sergeichuk, and N. Shaked-Monderer, “Solution of linear matrix equations in a $^{\ast\,\!}$congruence class,” Electronic Journal of Linear Algebra, vol. 13, pp. 153–156, 2005.
  • G. K. Hua, X. Hu, and L. Zhang, “A new iterative method for the matrix equation $AX=B$,” Applied Mathematics and Computation, vol. 15, pp. 1434–1441, 2007.
  • A. D. Porter, “Solvability of the matrix equation $AX=B$,” Linear Algebra and Its Applications, vol. 13, no. 3, pp. 177–184, 1976.
  • Q.-W. Wang and J. Yu, “On the generalized bi (skew-) symmetric solutions of a linear matrix equation and its procrust problems,” Applied Mathematics and Computation, vol. 219, no. 19, pp. 9872–9884, 2013.
  • Y. S. Hanna, “On the solutions of tridiagonal linear systems,” Applied Mathematics and Computation, vol. 189, no. 2, pp. 2011–2016, 2007.
  • C.-Z. Dong, Q.-W. Wang, and Y.-P. Zhang, “On the Hermitian \emphR-conjugate solution of a system of matrix equations,” Journal of Applied Mathematics, vol. 2012, Article ID 398085, 14 pages, 2012.
  • W. F. Trench, “Characterization and properties of matrices with generalized symmetry or skew symmetry,” Linear Algebra and Its Applications, vol. 377, pp. 207–218, 2004.
  • T. Huckle, S. Serra-Capizzano, and C. Tablino-Possio, “Preconditioning strategies for Hermitian indefinite Toeplitz linear systems,” SIAM Journal on Scientific Computing, vol. 25, no. 5, pp. 1633–1654, 2004.
  • R. H. Chan, A. M. Yip, and M. K. Ng, “The best circulant preconditioners for Hermitian Toeplitz systems,” SIAM Journal on Numerical Analysis, vol. 38, no. 3, pp. 876–896, 2000.
  • A. Lee, “Centro-Hermitian and skew-centro-Hermitian matrices,” Linear Algebra and Its Applications, vol. 29, pp. 205–210, 1980.
  • A. V. Oppenheim, Applications of Digital Signal Processing, Prentice-Hall, Englewood Cliffs, Calif, USA, 1978.
  • M. K. Ng, R. J. Plemmons, and F. Pimentel, “A new approach to constrained total least squares image restoration,” Linear Algebra and Its Applications, vol. 316, no. 1–3, pp. 237–258, 2000.
  • R. Kouassi, P. Gouton, and M. Paindavoine, “Approximation of the Karhunen-Loève tranformation and its application to colour images,” Signal Processing: Image Commu, vol. 16, pp. 541–551, 2001.
  • Y.-B. Deng, X.-Y. Hu, and L. Zhang, “Least squares solution of $BX{A}^{T}=T$ over symmetric, skew-symmetric, and positive semidefinite \emphX,” SIAM Journal on Matrix Analysis and Applications, vol. 25, no. 2, pp. 486–494, 2003.