RANKS OF SUBMATRICES IN A GENERAL SOLUTION TO A QUATERNION SYSTEM WITH APPLICATIONS

- Journal title : Bulletin of the Korean Mathematical Society
- Volume 48, Issue 5, 2011, pp.969-990
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/BKMS.2011.48.5.969

Title & Authors

RANKS OF SUBMATRICES IN A GENERAL SOLUTION TO A QUATERNION SYSTEM WITH APPLICATIONS

Zhang, Hua-Sheng; Wang, Qing-Wen;

Zhang, Hua-Sheng; Wang, Qing-Wen;

Abstract

Assume that X, partitioned into block form, is a solution of the system of quaternion matrix equations = . We in this paper give the maximal and minimal ranks of the submatrices in X, and establish necessary and sufficient conditions for the submatrices to be zero, unique as well as independent. As applications, we consider the common inner inverse G, partitioned into block form, of two quaternion matrices M and N. We present the formulas of the maximal and minimal ranks of the submatrices of G, and describe the properties of the submatrices of G as well. The findings of this paper generalize some known results in the literature.

Keywords

matrix equation;minimal rank;maximal rank;generalized inverse;quaternion matrix;partitioned matrix;

Language

English

Cited by

References

1.

M. Dehghan and M. Hajarian, An iterative algorithm for solving a pair of matrix equa- tions AY B = E, CY D = F over generalized centro-symmetric matrices, Comput. Math. Appl. 56 (2008), no. 12, 3246-3260.

2.

T. W. Hungerford, Algebra, Spring-Verlag Inc, New York, 1980.

3.

Y. Liu, Some properties of submatrices in a solution to the matrix equation AX = C, XB = D with applications, J. Appl. Math. Comput. 31 (2008), 71-80.

4.

S. K. Mitra, Common solutions to a pair of linear matrix equations A1XB1 = C1 and $A_2{\times}B_2\;=\;C_2$ , Proc. Cambridge Philos. Soc. 74 (1973), 213-216.

5.

S. K. Mitra, A pair of simultaneous linear matrix equations and a matrix programming problem, Linear Algebra Appl. 131 (1990), 107-123.

6.

A. B. Ozguler and N. Akar, A common solution to a pair of linear matrix equations over a principle domain, Linear Algebra Appl. 144 (1991), 85-99.

7.

Y. Tian, Some properties of submatrices in a solution to the matrix equation AXB = C with applications, J. Franklin Inst. 346 (2009), no. 6, 557-569.

8.

Y. Tian, The solvability of two linear matrix equations, Linear and Multilinear Algebra 48 (2000), no. 2, 123-147.

9.

Y. Tian, Upper and lower bounds for ranks of matrix expressions using generalized in- verses, Linear Algebra Appl. 355 (2002), 187-214.

10.

Y. Tian, The minimal rank completion of a $3{\times}3$ partial block matrix, Linear and Mul- tilinear Algebra 50 (2002), 125-131.

11.

F. Uhlig, On the matrix equation AX = B with applications to the generators of con- trollability matrix, Linear Algebra Appl. 85 (1987), 203-209.

12.

J. Van der Woude, Almost noninteracting control by measurement feedback, Systems Control Lett. 9 (1987), no. 1, 7-16.

13.

Q. W. Wang, The decomposition of pairwise matrices and matrix equations over an arbitrary skew eld, Acta Math. Sinica 39 (1996), no. 3, 396-403.

14.

Q. W. Wang, A system of matrix equations and a linear matrix equation over arbitrary reg- ular rings with identity, Linear Algebra Appl. 384 (2004), 43-54.