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Ranks and the least-norm of the general solution to a system of quaternion matrix equations. (English) Zbl 1158.15010
The authors consider the system of linear quaternion matrix equations ${A}_{1}{X}_{1}={C}_{1}$, ${A}_{2}{X}_{2}={C}_{2}$, ${A}_{3}{X}_{1}{B}_{1}+{A}_{4}{X}_{2}{B}_{2}={C}_{3}$ which is presumed consistent. They establish a new expression of its general solution; the system has been investigated recently by Q.-W. Wang, H.-X. Chang and C.-Y. Lin [Appl. Math. Comput. 195, No. 2, 721–732 (2008; Zbl 1149.15011)]. The authors derive the minimal and maximal ranks and the least-norm of the general solution to the system. Some previously known results are special cases of the ones in this paper.
##### MSC:
 15A24 Matrix equations and identities 15A33 Matrices over special rings 15A09 Matrix inversion, generalized inverses 15A03 Vector spaces, linear dependence, rank
##### References:
 [1] Hungerford, T. W.: Algebra, (1980) [2] Wang, Q. W.; Chang, H. X.; Lin, C. Y.: P-$\left(skew\right)$symmetric common solutions to a pair of quaternion matrix equations, Appl. math. Comput. 195, 721-732 (2008) · Zbl 1149.15011 · doi:10.1016/j.amc.2007.05.021 [3] Mitra, S. K.: The matrix equations AX=C, XB=D, Linear algebra appl. 59, 171-181 (1984) · Zbl 0543.15011 · doi:10.1016/0024-3795(84)90166-6 [4] Mitra, S. K.: A pair of simultaneous linear matrix equations A1XB1=C1, A2XB2=C2 and a programming problem, Linear algebra appl. 131, 107-123 (1990) · Zbl 0712.15010 · doi:10.1016/0024-3795(90)90377-O [5] Uhlig, F.: On the matrix equation AX=B with applications to the generators of controllability matrix, Linear algebra appl. 85, 203-209 (1987) · Zbl 0612.15006 · doi:10.1016/0024-3795(87)90217-5 [6] Wang, Q. W.; Wu, Z. C.; Lin, C. Y.: Extremal ranks of a quaternion matrix expression subject to consistent systems of quaternion matrix equations with applications, Appl. math. Comput. 182, 1755-1764 (2006) · Zbl 1108.15014 · doi:10.1016/j.amc.2006.06.012 [7] Wang, Q. W.; Song, G. J.; Lin, C. Y.: Extreme ranks of the solution to a consistent system of linear quaternion matrix equations with an application, Appl. math. Comput. 189, 1517-1532 (2007) · Zbl 1124.15010 · doi:10.1016/j.amc.2006.12.039 [8] Wang, Q. W.; Yu, S. W.; Lin, C. Y.: Extreme ranks of a linear quaternion matrix expression subject to triple quaternion matrix equations with applications, Appl. math. Comput. 195, 733-744 (2008) · Zbl 1149.15012 · doi:10.1016/j.amc.2007.05.018 [9] Liu, Y. H.: Ranks of solutions of the linear matrix equation AX+YB=C, Comput. math. Appl. 52, 861-872 (2006) · Zbl 1129.15009 · doi:10.1016/j.camwa.2006.05.011 [10] Tian, Y.: Ranks of solutions of the matrix equation AXB=C, Linear and multilinear algebra 51, No. 2, 111-125 (2003) · Zbl 1040.15003 · doi:10.1080/0308108031000114631 [11] Tian, Y.: Ranks and independence of solutions of the matrix equation AXB+CYD=M, Acta math. Univ. comenianae 1, 75-84 (2006) · Zbl 1164.15321 [12] Lin, C. Y.; Wang, Q. W.: The minimal and maximal ranks of the general solution to a system of matrix equations over an arbitrary division ring, Math. sci. Res. J. 10, No. 3, 57-65 (2006) · Zbl 1142.15302 [13] Q.W. Wang, G.J. Song, Maximal and minimal ranks of the common solution of some linear matrix equations over an arbitrary division ring, Algebra Colloquium, in press. · Zbl 1176.15020 [14] Tian, Y.: Upper and lower bounds for ranks of matrix expressions using generalized inverses, Linear algebra appl. 355, 187-214 (2002) · Zbl 1016.15003 · doi:10.1016/S0024-3795(02)00345-2 [15] Tian, Y.; Cheng, S.: The maximal and minimal ranks of A-BXC with applications, New York J. Math. 9, 345-362 (2003) · Zbl 1036.15004 · doi:emis:journals/NYJM/j/2003/9-18nf.htm [16] Tian, Y.: Solvability of two linear matrix equations, Linear and multilinear algebra 48, 123-147 (2000) · Zbl 0970.15005 · doi:10.1080/03081080008818664 [17] Wang, Q. W.; Chang, H. X.; Ning, Q.: The common solution to six quaternion matrix equations with applications, Appl. math. Comput. 198, 209-226 (2008) · Zbl 1141.15016 · doi:10.1016/j.amc.2007.08.091 [18] Wang, Q. W.: A system of matrix equations and a linear matrix equation over arbitrary regular rings with identity, Linear algebra appl. 384, 43-54 (2004) · Zbl 1058.15015 · doi:10.1016/j.laa.2003.12.039 [19] Marsaglia, G.; Styan, G. P. H.: Equalities and inequalities for ranks of matrices, Linear and multilinear algebra 2, 269-292 (1974) · Zbl 0297.15003 [20] Farenick, D. R.; Pidkowich, B. A. F.: The spectral theorem in quaternions, Linear algebra appl. 371, 75-102 (2003) · Zbl 1030.15015 · doi:10.1016/S0024-3795(03)00420-8 [21] Tian, Y.: Equalities and inequalities for traces of quaternionic matrices, Algebras groups geom. 19, No. 2, 181-193 (2002) · Zbl 1167.15308