##
**Least-norm of the general solution to some system of quaternion matrix equations and its determinantal representations.**
*(English)*
Zbl 1457.15017

Summary: We constitute some necessary and sufficient conditions for the system \(A_1 X_1 = C_1\), \(X_1 B_1 = C_2\), \(A_2 X_2 = C_3\), \(X_2 B_2 = C_4\), \(A_3 X_1 B_3 + A_4 X_2 B_4 = C_c\), to have a solution over the quaternion skew field in this paper. A novel expression of general solution to this system is also established when it has a solution. The least norm of the solution to this system is also researched in this article. Some former consequences can be regarded as particular cases of this article. Finally, we give determinantal representations (analogs of Cramer’s rule) of the least norm solution to the system using row-column noncommutative determinants. An algorithm and numerical examples are given to elaborate our results.

### MSC:

15A24 | Matrix equations and identities |

15B33 | Matrices over special rings (quaternions, finite fields, etc.) |

PDF
BibTeX
XML
Cite

\textit{A. Rehman} et al., Abstr. Appl. Anal. 2019, Article ID 9072690, 18 p. (2019; Zbl 1457.15017)

Full Text:
DOI

### References:

[1] | Hungerford, T., Algebra (1980), New York, NY, USA: Springer, New York, NY, USA |

[2] | Wang, Q., A system of matrix equations and a linear matrix equation over arbitrary regular rings with identity, Linear Algebra and its Applications, 384, 43-54 (2004) · Zbl 1058.15015 |

[3] | Zhang, F., Quaternions and matrices of quaternions, Linear Algebra and its Applications, 251, 21-57 (1997) · Zbl 0873.15008 |

[4] | Conway, J. H.; Smith, D. A., On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry (2003), Natick, MA, USA: A K Peters, Ltd., Natick, MA, USA · Zbl 1098.17001 |

[5] | Adler, S. L., Quaternionic Quantum Mechanics and Quantum Field (1995), New York, NY, USA: Oxford University Press, New York, NY, USA · Zbl 0885.00019 |

[6] | de Leo, S.; Scolarici, G., Right eigenvalue equation in quaternionic quantum mechanics, Journal of Physics A: Mathematical and Theoretical, 33, 15, 2971-2995 (2000) · Zbl 0954.81008 |

[7] | Took, C. C.; Mandic, D. P., Augmented second-order statistics of quaternion random signals, Signal Processing, 91, 2, 214-224 (2011) · Zbl 1203.94057 |

[8] | Took, C. C.; Mandic, D. P., The quaternion LMS algorithm for adaptive filtering of hypercomplex processes, IEEE Transactions on Signal Processing, 57, 4, 1316-1327 (2009) · Zbl 1391.93261 |

[9] | Took, C. C.; Mandic, D. P., A quaternion widely linear adaptive filter, IEEE Transactions on Signal Processing, 58, 8, 4427-4431 (2010) · Zbl 1392.94488 |

[10] | Took, C. C.; Mandic, D. P.; Zhang, F., On the unitary diagonalisation of a special class of quaternion matrices, Applied Mathematics Letters, 24, 11, 1806-1809 (2011) · Zbl 1388.15009 |

[11] | Shahzad, A.; Jones, B. L.; Kerrigan, E. C.; Constantinides, G. A., An efficient algorithm for the solution of a coupled Sylvester equation appearing in descriptor systems, Automatica, 47, 1, 244-248 (2011) · Zbl 1210.65147 |

[12] | Syrmos, V. L.; Lewis, F. L., Coupled and constrained sylvester equations in system design, Circuits, Systems and Signal Processing, 13, 6, 663-694 (1994) · Zbl 0864.93050 |

[13] | Varga, A., Robust pole assignment via Sylvester equation based state feedback parametrization, Proceedings of the CACSD. Conference Proceedings. IEEE International Symposium on Computer-Aided Control Systems Design |

[14] | Syrmos, V.; Lewis, F., Output feedback eigenstructure assignment using two Sylvester equations, IEEE Transactions on Automatic Control, 38, 3, 495-499 (1993) · Zbl 0791.93035 |

[15] | Li, R., A bound on the solution to a structured sylvester equation with an application to relative perturbation theory, SIAM Journal on Matrix Analysis and Applications, 21, 2, 440-445 (2000) · Zbl 0964.15018 |

[16] | Darouach, M., Solution to Sylvester equation associated to linear descriptor systems, Systems & Control Letters, 55, 10, 835-838 (2006) · Zbl 1100.93028 |

[17] | Zhang, Y.; Jiang, D.; Wang, J., A recurrent neural network for solving sylvester equation with time-varying coefficients, IEEE Transactions on Neural Networks and Learning Systems, 13, 5, 1053-1063 (2002) |

[18] | De Terán, F.; Dopico, F. M., The solution of the equation \(XA + A X^T = 0\) and its application to the theory of orbits, Linear Algebra and its Applications, 434, 1, 44-67 (2011) · Zbl 1204.15022 |

[19] | Dehghan, M.; Hajarian, M., Efficient iterative method for solving the second-order Sylvester matrix equation \(E V F^2 - A V F - C V = B W\), IET Control Theory & Applications, 3, 10, 1401-1408 (2009) |

[20] | Ding, F.; Chen, T., Gradient based iterative algorithms for solving a class of matrix equations, IEEE Transactions on Automatic Control, 50, 8, 1216-1221 (2005) · Zbl 1365.65083 |

[21] | Dmytryshyn, A.; Futorny, V.; Klymchuk, T.; Sergeichuk, V. V., Generalization of Roth’s solvability criteria to systems of matrix equations, Linear Algebra and its Applications, 527, 294-302 (2017) · Zbl 1365.15020 |

[22] | He, Z.; Wang, Q., A real quaternion matrix equation with applications, Linear and Multilinear Algebra, 61, 6, 725-740 (2013) · Zbl 1317.15016 |

[23] | He, Z.-H.; Wang, Q.-W., The \(η\)-bihermitian solution to a system of real quaternion matrix equations, Linear and Multilinear Algebra, 62, 11, 1509-1528 (2014) · Zbl 1307.15019 |

[24] | Kyrchei, I., Explicit representation formulas for the minimum norm least squares solutions of some quaternion matrix equations, Linear Algebra and its Applications, 438, 1, 136-152 (2013) · Zbl 1255.15022 |

[25] | Kyrchei, I., Explicit formulas for determinantal representations of the Drazin inverse solutions of some matrix and differential matrix equations, Applied Mathematics and Computation, 219, 14, 7632-7644 (2013) · Zbl 1291.15040 |

[26] | Rehman, A.; Wang, Q., A system of matrix equations with five variables, Applied Mathematics and Computation, 271, 805-819 (2015) · Zbl 1410.15033 |

[27] | Rehman, A.; Wang, Q.-W.; Ali, I.; Akram, M.; Ahmad, M. O., A constraint system of generalized Sylvester quaternion matrix equations, Advances in Applied Clifford Algebras (AACA), 27, 4, 3183-3196 (2017) · Zbl 1386.15035 |

[28] | Rehman, A.; Wang, Q.-W.; He, Z.-H., Solution to a system of real quaternion matrix equations encompassing \(\eta \)-Hermicity, Applied Mathematics and Computation, 265, 945-957 (2015) · Zbl 1410.15034 |

[29] | Rehman, A.; Akram, M., Optimization of a nonlinear Hermitian matrix expression with application, Filomat, 31, 9, 2805-2819 (2017) · Zbl 1488.15067 |

[30] | Wang, Q.-W.; Qin, F.; Lin, C.-Y., The common solution to matrix equations over a regular ring with applications, Indian Journal of Pure and Applied Mathematics, 36, 12, 655-672 (2005) · Zbl 1104.15014 |

[31] | Wang, Q.-W.; Rehman, A.; He, Z.-H.; Zhang, Y., Constraint generalized Sylvester matrix equations, Automatica, 69, 60-64 (2016) · Zbl 1338.93174 |

[32] | Bao, Y., Least-norm and extremal ranks of the least square solution to the quaternion matrix equation AXB=C subject to two equations, Algebra Colloquium, 21, 03, 449-460 (2014) · Zbl 1303.15018 |

[33] | Wang, Q.; Chang, H.; Lin, C., P-(skew)symmetric common solutions to a pair of quaternion matrix equations, Applied Mathematics and Computation, 195, 2, 721-732 (2008) · Zbl 1149.15011 |

[34] | Li, H.; Gao, Z.; Zhao, D., Least squares solutions of the matrix equation \(A X B + C Y D = E\) with the least norm for symmetric arrowhead matrices, Applied Mathematics and Computation, 226, 719-724 (2014) · Zbl 1354.15010 |

[35] | Wang, Q.-W.; van der Woude, J. W.; Chang, H.-X., A system of real quaternion matrix equations with applications, Linear Algebra and its Applications, 431, 12, 2291-2303 (2009) · Zbl 1180.15019 |

[36] | Peng, Y.-g.; Wang, X., A finite iterative algorithm for solving the least-norm generalized \((P, Q)\) reflexive solution of the matrix equations \(A_i X B_i = C_i\), Journal of Computational Analysis and Applications, 17, 3, 547-561 (2014) · Zbl 1291.65106 |

[37] | Yuan, S.; Liao, A., Least squares Hermitian solution of the complex matrix equation AXB+CXD=E with the least norm, Journal of The Franklin Institute, 351, 11, 4978-4997 (2014) · Zbl 1307.93175 |

[38] | Trench, W. F., Minimization problems for \((R, S)\)-symmetric and \((R, S)\)-skew symmetric matrices, Linear Algebra and its Applications, 389, 23-31 (2004) · Zbl 1059.15019 |

[39] | 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 |

[40] | Trench, W. F., Characterization and properties of \((R,S)\)-symmetric, \((R,S)\)-skew symmetric, and \((R,S)\)-conjugate matrices, SIAM Journal on Matrix Analysis and Applications, 26, 3, 748-757 (2005) · Zbl 1080.15022 |

[41] | Zhang, X., Characterization for the general solution to a system of matrix equations with quadruple variables, Applied Mathematics and Computation, 226, 274-287 (2014) · Zbl 1354.15011 |

[42] | Bapat, R. B.; Rao, K. P. S.; Prasad, K. M., Generalized inverses over integral domains, Linear Algebra and its Applications, 140, 181-196 (1990) · Zbl 0712.15004 |

[43] | Stanimirovic, P., General determinantal representation of pseudoinverses of matrices, Matematički Vesnik, 48, 1-9 (1996) · Zbl 0948.15005 |

[44] | Kyrchei, I. I.; Kyrchei, I. I., Cramer’s rule for generalized inverse solutions, Advances in Linear Algebra Research, 79-132 (2015), New York, NY, USA: Nova Science Publishers, New York, NY, USA |

[45] | Kyrchei, I. I., Cramer’s rule for quaternion systems of linear equations, Fundamentalnaya i Prikladnaya Matematika, 13, 4, 67-94 (2007) · Zbl 1157.15308 |

[46] | Kyrchei, I.; Baswell, A. R., The theory of the column and row determinants in a quaternion linear algebra, Advances in Mathematics Research, 15, 301-359 (2012), New York, NY, USA: Nova Science Publ, New York, NY, USA |

[47] | Kyrchei, I. I., Determinantal representations of the Moore-Penrose inverse over the quaternion skew field and corresponding Cramer’s rules, Linear and Multilinear Algebra, 59, 4, 413-431 (2011) · Zbl 1220.15007 |

[48] | Kyrchei, I. I., Explicit determinantal representation formulas of W-weighted drazin inverse solutions of some matrix equations over the quaternion skew field, Mathematical Problems in Engineering, 2016 (2016) · Zbl 1400.15009 |

[49] | Kyrchei, I., Determinantal representations of the Drazin inverse over the quaternion skew field with applications to some matrix equations, Applied Mathematics and Computation, 238, 193-207 (2014) · Zbl 1334.15012 |

[50] | Kyrchei, I., Determinantal representations of the W-weighted Drazin inverse over the quaternion skew field, Applied Mathematics and Computation, 264, 453-465 (2015) · Zbl 1410.15009 |

[51] | Kyrchei, I. I., Explicit determinantal representation formulas for the solution of the two-sided restricted quaternionic matrix equation, Applied Mathematics and Computation, 58, 1-2, 335-365 (2018) · Zbl 1396.15014 |

[52] | Kyrchei, I.; Griffin, S., Determinantal representations of the Drazin and w-weighted Drazin inverses over the quaternion skew field with applications, Quaternions: Theory and Applications, 201-275 (2017), New York, NY, USA: Nova Science Publishers, New York, NY, USA |

[53] | Kyrchei, I., Weighted singular value decomposition and determinantal representations of the quaternion weighted Moore-Penrose inverse, Applied Mathematics and Computation, 309, 1-16 (2017) · Zbl 1411.15025 |

[54] | Kyrchei, I. I.; Baswell, A. R., Determinantal representations of the quaternion weighted moore-penrose inverse and its applications, Advances in Mathematics Research, 23, 35-96 (2017), New York, NY, USA: Nova Science Publishers, New York, NY, USA |

[55] | Song, G.-J.; Wang, Q.-W.; Chang, H.-X., Cramer rule for the unique solution of restricted matrix equations over the quaternion skew field, Computers & Mathematics with Applications, 61, 6, 1576-1589 (2011) · Zbl 1217.15020 |

[56] | Song, G.-J.; Dong, C.-Z., New results on condensed Cramer’s rule for the general solution to some restricted quaternion matrix equations, Applied Mathematics and Computation, 53, 1-2, 321-341 (2017) · Zbl 1360.65129 |

[57] | Song, G.-J.; Wang, Q.-W., Condensed Cramer rule for some restricted quaternion linear equations, Applied Mathematics and Computation, 218, 7, 3110-3121 (2011) · Zbl 1248.15016 |

[58] | Song, G.-J.; Wang, Q.-W.; Yu, S.-W., Cramer’s rule for a system of quaternion matrix equations with applications, Applied Mathematics and Computation, 336, 490-499 (2018) · Zbl 1427.15019 |

[59] | Kyrchei, I., Determinantal representations of solutions to systems of quaternion matrix equations, Advances in Applied Clifford Algebras (AACA), 28, 1, article 23 (2018) · Zbl 1391.15057 |

[60] | Kyrchei, I. I., Determinantal representations of solutions and hermitian solutions to some system of two-sided quaternion matrix equations, Journal of Mathematics, 2018 (2018) · Zbl 1487.15017 |

[61] | Kyrchei, I. I., Cramer’s rules for Sylvester quaternion matrix equation and its special cases, Advances in Applied Clifford Algebras (AACA), 28, 5, 90 (2018) · Zbl 1404.15013 |

[62] | Kyrchei, I. I., Determinantal representations of general and (Skew-)Hermitian solutions to the generalized sylvester-type quaternion matrix equation, Abstract and Applied Analysis, 2019 (2019) · Zbl 1474.15041 |

[63] | Marsaglia, G.; Styan, G. P. H., Equalities and inequalities for ranks of matrices, Linear and Multilinear Algebra, 2, 269-292 (1974) · Zbl 0297.15003 |

[64] | Wang, Q.; Li, C., Ranks and the least-norm of the general solution to a system of quaternion matrix equations, Linear Algebra and its Applications, 430, 5-6, 1626-1640 (2009) · Zbl 1158.15010 |

[65] | Wang, Q. W.; Chang, H. X.; Ning, Q., The common solution to six quaternion matrix equations with applications, Applied Mathematics and Computation, 198, 1, 209-226 (2008) · Zbl 1141.15016 |

[66] | Tian, Y., The solvability of two linear matrix equations, Linear and Multilinear Algebra, 48, 2, 123-147 (2000) · Zbl 0970.15005 |

[67] | Wang, Q.; Wu, Z.; Lin, C., Extremal ranks of a quaternion matrix expression subject to consistent systems of quaternion matrix equations with applications, Applied Mathematics and Computation, 182, 2, 1755-1764 (2006) · Zbl 1108.15014 |

[68] | Dieudonne, J., Les determinantes sur un corps non-commutatif, Bulletin de la Société Mathématique de France, 71, 27-45 (1943) · Zbl 0028.33904 |

[69] | Dyson, F. J., Quaternion determinants Helvetica, Physica Acta, 45, 289-302 (1972) |

[70] | Gelfand, I.; Gelfand, S.; Retakh, V.; Wilson, R. L., Quasideterminants, Advances in Mathematics, 193, 1, 56-141 (2005) · Zbl 1079.15007 |

[71] | Cao, W., Solvability of a quaternion matrix equation, Applied Mathematics-A Journal of Chinese Universities, 17, 4, 490-498 (2002) · Zbl 1020.15014 |

[72] | Farenick, D. R.; Pidkowich, B. A., The spectral theorem in quaternions, Linear Algebra and its Applications, 371, 75-102 (2003) · Zbl 1030.15015 |

[73] | Tian, Y., Equalities and inequalities for traces of quaternionic matrices, Algebras, Groups and Geometries, 19, 2, 181-193 (2002) · Zbl 1167.15308 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.