# Ranks of the common solution to some quaternion matrix equations with applications

Document Type : Research Paper

Authors

1 Shanghai University

2 East China University of Science and Technology

Abstract

We derive the formulas of the maximal and
minimal ranks of four real matrices $X_{1},X_{2},X_{3}$ and $X_{4}$
in common solution $X=X_{1}+X_{2}i+X_{3}j+X_{4}k$ to quaternion
matrix equations $A_{1}X=C_{1},XB_{2}=C_{2},A_{3}XB_{3}=C_{3}$. As
applications, we establish necessary and sufficient conditions for\
the existence of the common real and complex solutions to the matrix
equations. We give the expressions of such solutions to this system
when the solvability conditions are met. Moreover, we present
necessary and sufficient conditions for the existence of real and
complex solutions to the system of quaternion
matrix equations $A_{1}X=C_{1},XB_{2}=C_{2},A_{3}XB_{3}=C_{3},A_{4}% XB_{4}=C_{4}$. The findings of this paper extend some known results
in the literature.

Keywords

### History

• Receive Date: 10 August 2010
• Revise Date: 15 September 2010
• Accept Date: 15 August 2010
• First Publish Date: 01 April 2012