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Articles

Least squares η-bi-Hermitian problems of the quaternion matrix equation (AXB,CXD) = (E,F)

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Pages 1849-1863 | Received 08 Oct 2013, Accepted 10 Oct 2014, Published online: 04 Nov 2014

References

  • Ell T, Sangwine SJ. Quaternion involutions and anti-involutions. Comput. Math. Appl. 2007;53:137–143.
  • Zhang FZ. Quaternions and matrices of quaternions. Linear Algebra Appl. 1997;251:21–57.
  • Horn RA, Zhang FZ. A generalization of the complex Autonne-Takagi factorization to quaternion matrices. Linear Multilinear Algebra. 2012;60:1239–1244.
  • Took CC, Mandic DP. Augmented second-order statistics of quaternion random signals. Signal Process. 2011;91:214–224.
  • Took CC, Mandic DP, Zhang FZ. On the unitary diagonalisation of a special class of quaternion matrices. Appl. Math. Lett. 2011;24:1806–1809.
  • Peng ZY, Hu XY, Zhang L. The inverse problem of bisymmetric matrices with a submatrix constraint. Numer. Linear Algebra Appl. 2004;11:59–73.
  • Xie DX, Hu XY, Zhang L. The solvability conditions for inverse eigenvalue problem of anti-bisymmetric matrices. J. Comput. Math. 2002;20:245–256.
  • Xie DX, Zhang L, Hu XY. The solvability conditions for the inverse problem of bisymmetric nonnegative definite matrices. J. Comput. Math. 2000;18:597–608.
  • Yuan SF, Wang QW, Xiong ZP. Linear parameterized inverse eigenvalue problem of bisymmetric matrices. Linear Algebra Appl. 2013;439:1990–2007.
  • Dai H, Lancaster P. Linear matrix equations from an inverse problem of vibration theory. Linear Algebra Appl. 1996;246:31–47.
  • Dehghan M, Hajarian M. On the reflexive solutions of the matrix equation AXB + CYD = E. Bull. Korean Math. Soc. 2009;46:511–519.
  • Deng YB, Hu XY. On solutions of matrix equation AXAT + BYBT = C. J. Comput. Math. 2005;23:17–26.
  • He ZH, Wang QW. The η-bi-Hermitian solution to a system of real quaternion matrix equations. Linear Multilinear Algebra. 2014;62:1509–1528.
  • Huang LP. The matrix equation AXB – CXD = E over the quaternion field. Linear Algebra Appl. 1996;234:197–208.
  • Huang LP, Liu JZ. The extension of Roth’s theorem for matrix equations over a ring. Linear Algebra Appl. 1997;259:229–235.
  • Jiang TS, Wei MS. On a solution of the quaternion matrix equation X – AX͂B = C and its application. Acta Math. Sin. 2005;21:483–490.
  • Lei Y, Liao AP, Zhang L. Minimization problem for symmetric orthogonal anti-symmetric matrices. J. Comput. Math. 2007;25:211–220.
  • Li YT, Wu WJ. Symmetric and skew-antisymmetric solutions to systems of real quaternion matrix equations. Comput. Math. Applic. 2008;55:1142–1147.
  • Liao AP, Bai ZZ, Lei Y. Best approximate solution of matrix equation AXB + CYD = E. SIAM J. Matrix Anal. Appl. 2006;27:675–688.
  • Liao AP, Lei Y, Yuan SF. The matrix nearness problem for symmetric matrices associated with the matrix equation [AT XA, BT XB] = [C, D]. Linear Algebra Appl. 2006;418:939–954.
  • Liao AP, Lei Y. Least-squares solution with the minimum-norm for the matrix equation (AXB, GXH) = (C, D). Comput. Math. Applic. 2005;50:539–549.
  • Liu YH, Jiang TS, Wei MS. The singular value decomposition of quaternion matrix with application. Numer. Math. J. Chinese Univ. 2003;25:321–328 . Chinese.
  • Magnus JR. L-structured matrices and linear matrix equations. Linear Multilinear Algebra. 1983;14:67–88.
  • Özgüler A. The equation AXB + CYD = E over a principle ideal domain. SIAM J. Matrix Anal. Appl. 1991;12:581–591.
  • Shim S, Chen Y. Least squares solution of matrix equation AXB* = CYD* = E. SIAM J. Matrix Anal. Appl. 2003;24:802–808.
  • Song CQ, Chen GL, Liu QB. Explicit solutions to the quaternion matrix equations X – AXF = C and X – AXF = C. Int. J. Comput. Math. 2012;89:890–900.
  • Xu GP, Wei MS, Zheng DS. On solutions of matrix equation AXB+ CYD = F. Linear Algebra Appl. 1998;279:93–109.
  • Yu L, Zhang KY, Shi ZK. The anti-symmetric ortho-symmetric solution of a linear matrix equation and its optimal approximation. J. Appl. Math. Comput. 2008;27:97–106.
  • Yuan SF, Liao AP, Lei Y. Least squares Hermitian solution of the matrix equation (AXB, CXD) = (E, F) with the least norm over the skew field of quaternions. Math. Comput. Model. 2008;48:91–100.
  • Yuan SF, Liao AP. Least squares solution of quaternion matrix equation X – AX̂B = C with the least norm. Linear Multilinear Algebra. 2011;59:985–998.
  • Yuan SF, Wang QW, Zhang X. Least-squares problem for the quaternion matrix equation AXB + CYD = E over different constrained matrices. Int. J. Comput. Math. 2012;90:565–576.
  • Yuan YX, Dai H. The least squares solution with the minimum norm of matrix equation AT XB + BTXT A= D. Numer. Math. J. Chinese Univ. 2005;27:232–238 . Chinese.
  • Wang QW. Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations. Comput. Math. Appl. 2005;49:641–650.
  • Wang QW, Woude vander JW, Chang HX. A system of real quaternion matrix equations with applications. Linear Algebra Appl. 2009;431:2291–2303.
  • Wang QW, Zhang F. The reflexive re-nonnegative definite solution to a quaternion matrix equation. Electron. J. Linear Algebra. 2008;17:88–101.
  • Zhang X. The general Hermitian nonnegative-definite and positive-definite solutions to the matrix equation GXG* HXH* = C. J. Appl. Math. Comput. 2004;14:51–67.
  • Ben-Israel A, Greville TNE. Generalized inverses: theory and applications. New York (NY): Wiley; 1974.

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