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Numerical solutions of AXB = C for mirrorsymmetric matrix X under a specified submatrix constraint

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Abstract

Mirrorsymmetric matrices, which are the iteraction matrices of mirrorsymmetric structures, have important application in studying odd/even-mode decomposition of symmetric multiconductor transmission lines (MTL). In this paper we present an efficient algorithm for minimizing \({\|AXB-C\|}\) where \({\|\cdot\|}\) is the Frobenius norm, \({A\in \mathbb{R}^{m\times n}}\), \({B\in \mathbb{R}^{n\times s}}\), \({C\in \mathbb{R}^{m\times s}}\) and \({X\in \mathbb{R}^{n\times n}}\) is mirrorsymmetric with a specified central submatrix [x ij ]r≤i, j≤n-r. Our algorithm produces a suitable X such that AXB = C in finitely many steps, if such an X exists. We show that the algorithm is stable any case, and we give results of numerical experiments that support this claim.

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References

  1. Datta L, Morgera S (1989) On the reducibility of centrosymmetric matrices-applications in engineering problems. Circuits Syst Signal Process 8: 71–96

    Article  MATH  MathSciNet  Google Scholar 

  2. Li GL, Feng ZH (2002) Mirrorsymetric matrices, their basic properties, and an application on odd/even-mode decomposition of symmetric multiconductor transmission lines. SIAM J Matrix Anal Appl 24: 78–90

    Article  MATH  MathSciNet  Google Scholar 

  3. Li GL, Feng ZH (2003) Mirror-transformations of matrices and their application on odd/even modal decomposition of mirror-symmetric multiconductor transmission line equations. IEEE Trans Adv Pack 26: 172–181

    Article  Google Scholar 

  4. Li GL, Feng ZH (2002) Mirror-symmetric matrices and their application. Tsinghua Sci Technol 7: 602–607

    MATH  MathSciNet  Google Scholar 

  5. Weaver J (1985) Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, and eigenvectors. Am Math Mon 92: 711–717

    Article  MATH  MathSciNet  Google Scholar 

  6. Cantoni A, Butler P (1976) Eigenvalues and eigenvectors of symmetric centrosymmetric matrices. Linear Algebra Appl 13: 275–288

    Article  MATH  MathSciNet  Google Scholar 

  7. Peng ZY, Hu XY, Zhang L (2004) The inverse problem of symmetric centrosymmetric matrices with a submatrix constraint. Linear Algebra Appl 11: 59–73

    Article  MATH  MathSciNet  Google Scholar 

  8. Bai ZJ (2005) The inverse eigenproblem of centrosymmetric matrices with a submatrix constraint and its approximation. SIAM J Matrix Anal Appl 26: 1100–1114

    Article  MATH  MathSciNet  Google Scholar 

  9. Liao AP, Lei Y (2007) Least-squares solutions of matrix inverse problem for bi-symmetric matrices with a submatrix constraint. Linear Algebra Appl 14: 425–444

    Article  MATH  MathSciNet  Google Scholar 

  10. Deift P, Nanda T (1984) On the determination of a tridiagonal matrix from its spectrum and a submatrix. Linear Algebra Appl 60: 43–55

    Article  MATH  MathSciNet  Google Scholar 

  11. Peng ZY, Hu XY (2005) Constructing Jacobi matrix with prescribed ordered defective eigenpairs and a principal submatrix. J Comput Appl Math 175: 321–333

    Article  MATH  MathSciNet  Google Scholar 

  12. Yuan YX, Dai H (2007) Inverse problems for symmetric matrices with a submatrix constraint. Appl Numer Math 57: 646–656

    Article  MATH  MathSciNet  Google Scholar 

  13. Yuan YX, Dai H (2008) The nearness problems for symmetric matrix with a submatrix constraint. J Comput Appl Math 213: 224–231

    Article  MATH  MathSciNet  Google Scholar 

  14. Gong LS, Hu XY, Zhang L (2006) The expansion problem of anti-symmetric matrix under a linear constraint and the optimal approximation. J Comput Appl Math 197: 44–52

    Article  MATH  MathSciNet  Google Scholar 

  15. Yin QX (2004) Construction of real antisymmetric and bi-antisymmetric matrices with prescribed spectrum data. Linear Algebra Appl 389: 95–106

    Article  MATH  MathSciNet  Google Scholar 

  16. Lancaster P (1970) Explicit solutions of linear matrix equation. SIAM Rev 72: 544–566

    Article  MathSciNet  Google Scholar 

  17. Dai H, Lancaster P (1996) Linear matrix equations from an inverse problem of vibration theory. Linear Algebra Appl 246: 31–47

    Article  MATH  MathSciNet  Google Scholar 

  18. Beattie CA, Smith SW (1992) Optimal matrix approximants in structural identification. J Optim Theor Appl 74(1): 23–56

    Article  MATH  MathSciNet  Google Scholar 

  19. Zhao LJ, Hu XY, Zhang L (2008) Least squares solutions to AX = B for symmetric centrosymmetric matrices under a central principal submatrix constraint and the optimal approximation. Linear Algebra Appl 428: 871–880

    Article  MATH  MathSciNet  Google Scholar 

  20. Golub GH, Zha HY (1994) Perturbation analysis of the canonical correlation of matrix pairs. Linear Algebra Appl 210: 3–28

    Article  MATH  MathSciNet  Google Scholar 

  21. Paige CC (1986) Computing the generalized singular value decomposition. SIAM J Sci Stat Comput 7: 1126–1146

    Article  MATH  MathSciNet  Google Scholar 

  22. Lei Y, Liao AP (2007) A minimal residual algorithm for the inconsistent matrix equation AXB = C over symmetric matrices. Appl Math Comput 188: 499–513

    Article  MATH  MathSciNet  Google Scholar 

  23. Peng YX, Hu XY, Zhang L (2005) An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation AXB =  C. Appl Math Comput 160: 763–777

    MATH  MathSciNet  Google Scholar 

  24. Wang RS (2003) Functional analysis and optimization theory. Beljing Unlv of Aeronautics Astronautics Press, Beijing

    Google Scholar 

  25. Kelley CT (1995) Iterative methods for linear and nonlinear equations. SIAM, Philadelphia

    MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin, 541004, People’s Republic of China

    Jiao-Fen Li & Xue-Feng Duan

  2. College of Mathematics and Econometrics, Hunan University, Changsha, 410082, Hunan, People’s Republic of China

    Xi-Yan Hu & Lei Zhang

Authors
  1. Jiao-Fen Li
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  2. Xi-Yan Hu
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  3. Xue-Feng Duan
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  4. Lei Zhang
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Corresponding author

Correspondence to Jiao-Fen Li.

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Communicated by C.C. Douglas.

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Li, JF., Hu, XY., Duan, XF. et al. Numerical solutions of AXB = C for mirrorsymmetric matrix X under a specified submatrix constraint. Computing 90, 39–56 (2010). https://doi.org/10.1007/s00607-010-0109-9

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  • DOI: https://doi.org/10.1007/s00607-010-0109-9

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