Guang-Jing Song

Extreme ranks of the solution to a consistent system of linear quaternion matrix equations with an application

Abstract In this paper, we establish the maximal and minimal ranks of the solution to the consistent system of quaternion matrix equations A 1 X = C 1 , A 2 X = C 2 , A 3 XB 3 = C 3 and A 4 XB 4 = C 4 , which was investigated recently by Wang [Q.W. Wang, The general solution to a system of real quaternion matrix equations, Comput. Math. Appl. 49 (2005) 665–675]. Moreover, corresponding results on some special cases are presented. As an application, a necessary and sufficient condition for the invariance of the rank of the general solution to the system mentioned above is presented. Some previous known results can be regarded as the special cases of this paper.

A new solvable condition for a pair of generalized Sylvester equations

A necessary and sufficient condition is given for the quaternion matrix equations AiX + YB i = Ci (i =1 , 2) to have a pair of common solutions X and Y. As a consequence, the results partially answer a question posed by Y.H. Liu (Y.H. Liu, Ranks of solutions of the linear matrix equation AX + YB = C, Comput. Math. Appl., 52 (2006), pp. 861-872).

Condensed Cramer rule for some restricted quaternion linear equations

Abstract In this paper, we derive an alternative more condensed Cramer rule for the unique solution of some restricted left and right systems of quaternion linear equations. The findings of this paper extend some known results in the literature.

Maximal and Minimal Ranks of the Common Solution of Some Linear Matrix Equations over an Arbitrary Division Ring with Applications

We establish the formulas of the maximal and minimal ranks of the common solution of certain linear matrix equations A1X = C1, XB2 = C2, A3XB3 = C3 and A4XB4 = C4 over an arbitrary division ring. Corresponding results in some special cases are given. As an application, necessary and sufficient conditions for the invariance of the rank of the common solution mentioned above are presented. Some previously known results can be regarded as special cases of our results.

Determinantal representation of the generalized inverses over the quaternion skew field with applications

We first present some determinantal representations of one {1,5}-inverse of a quaternion matrix within the framework of a theory of the row and column determinants. As applications, we show some new explicit expressions of generalized inverses \(A_{r_{T_{1}, S_{1}}}^{( 2)}\), \(A_{l_{T_{2},S_{2}}}^{(2)}\) and \(A_{_{( T_{1},T_{2}) , ( S_{1},S_{2}) }}^{( 2) }\) over the quaternion skew field. Finally, we give the representations of the unique solution to some restricted left and right systems of quaternionic linear equations. The findings of this paper extend some known results in the literature.

Rank equalities related to the generalized inverse AT,S(2) with applications

Abstract A variety of rank formulas of some matrix expressions and certain partitioned matrices with respect to the generalized inverse A T , S ( 2 ) are established. Some necessary and sufficient conditions are given by using the rank formulas presented in this paper for two, three and four ordered matrices to be independent in the generalized inverse A T , S ( 2 ) . As special cases, necessary and sufficient conditions are derived for two, three and four ordered matrices to be independent in the weighted Moore–Penrose inverse and the Drazin inverse. Some known results can be regarded as the special cases of the results in this paper.

Cramer’s rule for a system of quaternion matrix equations with applications

In this paper, we investigate Cramer’s rule for the general solution to the system of quaternion matrix equations A1XB1=C1,A2XB2=C2,and Cramer’s rule for the general solution to the generalized Sylvester quaternion matrix equation AXB+CYD=E,respectively. As applications, we derive the determinantal expressions for the Hermitian solutions to some quaternion matrix equations. The findings of this paper extend some known results in the literature.

The block independence in the generalized inverse AT,S(2) for some ordered matrices and applications☆

Abstract In this paper, the definition of block independence in the generalized inverse A T , S ( 2 ) is firstly given, and then a necessary and sufficient condition for some ordered matrices to be block independent in the generalized inverse A T , S ( 2 ) is derived. As an application, a necessary and sufficient condition for A 1 + A 2 + ⋯ + A k T , S ( 2 ) = A 1 T 1 , S 1 ( 2 ) + A 2 T 2 , S 2 ( 2 ) + ⋯ + A k T k , S k ( 2 ) is proved. Moreover, some results are shown with respect to the Moore–Penrose inverse, the Weighted Moore–Penrose inverse and the Drazin inverse, respectively.