A Real Representation Method for Solving Yakubovich-j-Conjugate Quaternion Matrix Equation

Abstract

A new approach is presented for obtaining the solutions to Yakubovich-$j$-conjugate quaternion matrix equation $X-A\widehat{X}B=CY$ based on the real representation of a quaternion matrix. Compared to the existing results, there are no requirements on the coefficient matrix $A$. The closed form solution is established and the equivalent form of solution is given for this Yakubovich-$j$-conjugate quaternion matrix equation. Moreover, the existence of solution to complex conjugate matrix equation $X-A\overline{X}B=CY$ is also characterized and the solution is derived in an explicit form by means of real representation of a complex matrix. Actually, Yakubovich-conjugate matrix equation over complex field is a special case of Yakubovich-$j$-conjugate quaternion matrix equation $X-A\widehat{X}B=CY$. Numerical example shows the effectiveness of the proposed results.