Ranks and the least-norm of the general solution to a system of quaternion matrix equations

We in this paper first establish a new expression of the general solution to the consistent system of linear quaternion matrix equations A1X1=C1, A2X2=C2, A3X1B1+A4X2B2=C3, which was investigated recently [Q.W. Wang, H.X. Chang, C.Y. Lin, P-(skew)symmetric common solutions to a pair of quaternion matrix equations, Appl. Math. Comput. 195 (2008) 721–732], then derive the maximal and minimal ranks and the least-norm of the general solution to the system mentioned above. Some previous known results can be viewed as special cases of the results of this paper.

[1]  Sujit Kumar Mitra,et al.  A pair of simultaneous linear matrix equations A1XB1 = C1, A2XB2 = C2 and a matrix programming problem , 1990 .

[2]  Qing-Wen Wang,et al.  Extreme ranks of the solution to a consistent system of linear quaternion matrix equations with an application , 2007, Appl. Math. Comput..

[3]  Qing-Wen Wang,et al.  Extreme ranks of a linear quaternion matrix expression subject to triple quaternion matrix equations with applications , 2008, Appl. Math. Comput..

[4]  R. Tennant.  Algebra , 1941, Nature.

[5]  Yongge Tian,et al.  The maximal and minimal ranks of A − BXC with applications , 2003 .

[6]  Qing-Wen Wang,et al.  The common solution to six quaternion matrix equations with applications , 2008, Appl. Math. Comput..

[7]  Qingwen Wang,et al.  A System of Matrix Equations and a Linear Matrix Equation Over Arbitrary Regular Rings with Identity , 2003 .

[8]  Guang-Jing Song,et al.  Maximal and Minimal Ranks of the Common Solution of Some Linear Matrix Equations over an Arbitrary Division Ring with Applications , 2009 .

[9]  D. Farenick,et al.  The spectral theorem in quaternions , 2003 .

[10]  S. Mitra.  The matrix equations AX = C, XB = D , 1984 .

[11]  Yongge Tian.  Ranks of Solutions of the Matrix Equation AXB = C , 2003 .

[12]  Qing-Wen Wang,et al.  Extremal ranks of a quaternion matrix expression subject to consistent systems of quaternion matrix equations with applications , 2006, Appl. Math. Comput..

[13]  Qing-Wen Wang,et al.  P-(skew)symmetric common solutions to a pair of quaternion matrix equations , 2008, Appl. Math. Comput..

[14]  Yongge Tian.  The solvability of two linear matrix equations , 2000 .

[15]  Yongge Tian.  Upper and lower bounds for ranks of matrix expressions using generalized inverses , 2002 .

[16]  Yongge Tian.  RANKS AND INDEPENDENCE OF SOLUTIONS OF THE MATRIX EQUATION AXB + CY D = M , 2006 .

[17]  F. Uhlig.  On the matrix equation AX=B with applications to the generators of a controllability matrix , 1987 .

[18]  G. Styan,et al.  Equalities and Inequalities for Ranks of Matrices , 1974 .

[19]  Yong Hui Liu,et al.  Ranks of Solutions of the Linear Matrix Equation AX + YB = C , 2006, Comput. Math. Appl..