An $$n\times n $$ n × n quaternion matrix is said to be $$\eta $$ η -Hermitian if $$A=A^{\eta {*}}$$ A = A η ∗ , where $$A^{\eta {*}}=-\eta A^{*}\eta $$ A η ∗ = - η A ∗ η , $$\eta $$ η is one of the quaternion units i, j, k, and $$A^{*}$$ A ∗ is the conjugate transpose of A. In this paper, we investigate the generalized Sylvester quaternion matrix equation $$\begin{aligned} A_{1}X_{1}B_{1}+A_{2}X_{2}B_{2}+A_{3}X_{3}B_{3}=C. \end{aligned}$$ A 1 X 1 B 1 + A 2 X 2 B 2 + A 3 X 3 B 3 = C . We establish the necessary and sufficient conditions for the existence of a solution to this equation, and give an expression of the general solution to the equation when it is solvable. As an application, we derive the solvability conditions for the quaternion matrix equation $$\begin{aligned} A_{1}X_{1}A_{1}^{\eta *}+A_{2}X_{2}A_{2}^{\eta *}+A_{3}X_{3}A_{3} ^{\eta *}=C \end{aligned}$$ A 1 X 1 A 1 η ∗ + A 2 X 2 A 2 η ∗ + A 3 X 3 A 3 η ∗ = C to have an $$\eta $$ η -Hermitian solution as well as an expression of the $$\eta $$ η -Hermitian solution.
Advances in Applied Clifford Algebras – Springer Journals
Published: Apr 22, 2017
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