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Completion of Symmetric Orthogonal Matrices and Corresponding Matrix Equation Problem

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Abstract

The completion problem for symmetric orthogonal matrix and corresponding matrix equation problem are studied. We give some necessary and sufficient conditions of solvability and the expressions of solutions. In addition, in the solution set of the matrix equation problem, the expression of the optimal approximation matrix to a given matrix is obtained. Finally, the algorithms and numerical experiments are given for solving these problems.

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References

  1. Albert, A.: Conditions for positive and nonnegative definiteness in terms of pseudoinverses. SIAM J. Appl. Math. 17(2), 434–440 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  2. Fiedler, M., Markham, T.L.: Completing a matrix when certain entries of its inverse are specified. Linear Algebra Appl. 74(86), 225–237 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  3. Dai, H.: Completing a symmetric \(2 \times 2\) block matrix and its inverse. Linear Algebra Appl. 235, 235–245 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Meng, C.J., Hu, X.Y., Zhang, L., Uhlig, F.: Completion of an orthogonal projector and a related inverse mapping problem. Linear Algebra Appl. 403, 229–247 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Rubei, E.: On completions of symmetric and antisymmetric block diagonal partial matrices. Linear Algebra Appl. 439(10), 2971–2979 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Jameson, A., Kreindler, E.: Inverse problem of linear optimal control. SIAM J. Control 11(1), 1–19 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  7. Joseph, K.T.: Inverse eigenvalue problem in structural design. AIAA J. 30(12), 2890–2896 (2012)

    Article  MATH  Google Scholar 

  8. Yuan, S.F., Wang, Q.W., Duan, X.F.: On solutions of the quaternion matrix equation \(AX=B\) and their applications in color image restoration. Appl. Math. Comput. 221, 10–20 (2013)

    MathSciNet  MATH  Google Scholar 

  9. Li, Jh, Wang, D., Duan, J.A., He, H., Xia, Y., Zhu, W.H.: Structural design and control of a small-mrf damper under 50 n soft-landing applications. IEEE Trans. Ind. Inf. 11(3), 612–619 (2015)

    Article  Google Scholar 

  10. Li, J.H., Tian, W.Y., Liao, H.L., Zhou, C., Liu, X.H., Zhu, W.H.: The mathematical model & novel final test system for wafer-level packaging. IEEE Trans. Ind. Inf. 13(4), 1817–1824 (2017)

    Article  Google Scholar 

  11. Khatri, C.G., Mitra, S.K.: Hermitian and nonnegative definite solutions of linear matrix equations. SIAM J. Appl. Math. 31(4), 579–585 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  12. Don, F.J.H.: On the symmetric solutions of a linear matrix equation. Linear Algebra Appl. 93, 1–7 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chu, K.E.: Symmetric solutions of linear matrix equations by matrix decompositions. Linear Algebra Appl. 119, 35–50 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dai, H.: The best approximation by real symmetric matrices on the linear manifold. Math. Numer. Sin. 15(4), 478–488 (1993)

    MathSciNet  MATH  Google Scholar 

  15. Woodgate, K.G.: Least-squares solution of \(F=PG\) over positive semidefinite symmetric \(P\). Linear Algebra Appl. 245(3), 171C190 (1996)

    MathSciNet  MATH  Google Scholar 

  16. Peng, Z.Y., Hu, X.Y.: The reflexive and anti-reflexive solutions of the matrix equation \(AX=B\). Linear Algebra Appl. 375, 147–155 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhang, J.C., Zhou, S.Z., Hu, X.Y.: The \((P, Q)\) generalized reflexive and anti-reflexive solutions of the matrix equation \(AX=B\). Appl. Math. Comput. 209, 254–258 (2009)

    MathSciNet  MATH  Google Scholar 

  18. Wang, Q.W.: Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations. Comput. Math. Appl. 49(5), 641–650 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Zhao, L.J., Hu, X.Y., Zhang, L.: Least squares solutions to \(AX=B\) for bisymmetric matrices under a central principal submatrix constraint and the optimal approximation. Linear Algebra Appl. 428(4), 871–880 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wang, Q.W., Yu, J.: On the generalized bi (skew-) symmetric solutions of a linear matrix equation and its procrust problems. Appl. Math. Comput. 219, 9872–9884 (2013)

    MathSciNet  MATH  Google Scholar 

  21. Meng, C.J., Hu, X.Y., Zhang, L.: The skew-symmetric orthogonal solutions of the matrix equation \(AX=B\). Linear Algebra Appl. 402(1), 303–318 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Meng, C.J.: The theory and algorithm research for solving several classes of constrained matrix equation. Ph.D. Thesis, Hunan University, Changsha, pp. 35–53 (2004)

  23. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis, p. 149. Posts & Telecom Press, Beijing (2005)

    Google Scholar 

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Acknowledgements

This work was supported by Scientific Research Fund of Scientific and Technological Project of Changsha City, (Grant Nos. ZD1601077, K1705078).

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Authors and Affiliations

  1. College of Computer Engineering and Applied Mathematics, Changsha University, Hunan, China

    Wei Liu, Xiangyang Peng, Jinrong Shen & Baiyu Wang

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  1. Wei Liu
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  2. Xiangyang Peng
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  3. Jinrong Shen
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  4. Baiyu Wang
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Correspondence to Baiyu Wang.

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Liu, W., Peng, X., Shen, J. et al. Completion of Symmetric Orthogonal Matrices and Corresponding Matrix Equation Problem. Int. J. Appl. Comput. Math 4, 99 (2018). https://doi.org/10.1007/s40819-018-0530-x

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  • DOI: https://doi.org/10.1007/s40819-018-0530-x

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