## 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.)
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### References:

  Hungerford, T., Algebra (1980), New York, NY, USA: Springer, New York, NY, USA  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  Zhang, F., Quaternions and matrices of quaternions, Linear Algebra and its Applications, 251, 21-57 (1997) · Zbl 0873.15008  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  Adler, S. L., Quaternionic Quantum Mechanics and Quantum Field (1995), New York, NY, USA: Oxford University Press, New York, NY, USA · Zbl 0885.00019  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  Took, C. C.; Mandic, D. P., Augmented second-order statistics of quaternion random signals, Signal Processing, 91, 2, 214-224 (2011) · Zbl 1203.94057  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  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  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  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  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  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  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  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  Darouach, M., Solution to Sylvester equation associated to linear descriptor systems, Systems & Control Letters, 55, 10, 835-838 (2006) · Zbl 1100.93028  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)  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  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)  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  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  He, Z.; Wang, Q., A real quaternion matrix equation with applications, Linear and Multilinear Algebra, 61, 6, 725-740 (2013) · Zbl 1317.15016  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  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  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  Rehman, A.; Wang, Q., A system of matrix equations with five variables, Applied Mathematics and Computation, 271, 805-819 (2015) · Zbl 1410.15033  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  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  Rehman, A.; Akram, M., Optimization of a nonlinear Hermitian matrix expression with application, Filomat, 31, 9, 2805-2819 (2017) · Zbl 1488.15067  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  Wang, Q.-W.; Rehman, A.; He, Z.-H.; Zhang, Y., Constraint generalized Sylvester matrix equations, Automatica, 69, 60-64 (2016) · Zbl 1338.93174  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  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  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  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  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  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  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  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  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  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  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  Stanimirovic, P., General determinantal representation of pseudoinverses of matrices, Matematički Vesnik, 48, 1-9 (1996) · Zbl 0948.15005  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  Kyrchei, I. I., Cramer’s rule for quaternion systems of linear equations, Fundamentalnaya i Prikladnaya Matematika, 13, 4, 67-94 (2007) · Zbl 1157.15308  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  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  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  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  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  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  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  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  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  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  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  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  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  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  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  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  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  Marsaglia, G.; Styan, G. P. H., Equalities and inequalities for ranks of matrices, Linear and Multilinear Algebra, 2, 269-292 (1974) · Zbl 0297.15003  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  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  Tian, Y., The solvability of two linear matrix equations, Linear and Multilinear Algebra, 48, 2, 123-147 (2000) · Zbl 0970.15005  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  Dieudonne, J., Les determinantes sur un corps non-commutatif, Bulletin de la Société Mathématique de France, 71, 27-45 (1943) · Zbl 0028.33904  Dyson, F. J., Quaternion determinants Helvetica, Physica Acta, 45, 289-302 (1972)  Gelfand, I.; Gelfand, S.; Retakh, V.; Wilson, R. L., Quasideterminants, Advances in Mathematics, 193, 1, 56-141 (2005) · Zbl 1079.15007  Cao, W., Solvability of a quaternion matrix equation, Applied Mathematics-A Journal of Chinese Universities, 17, 4, 490-498 (2002) · Zbl 1020.15014  Farenick, D. R.; Pidkowich, B. A., The spectral theorem in quaternions, Linear Algebra and its Applications, 371, 75-102 (2003) · Zbl 1030.15015  Tian, Y., Equalities and inequalities for traces of quaternionic matrices, Algebras, Groups and Geometries, 19, 2, 181-193 (2002) · Zbl 1167.15308
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