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| Let $mathbb{H}^{m imes n}$ be the set of all$m imes n$ matrices over the real quaternion algebra $mathbb{H}%=ig{a_{0}+a_{1}i+a_{2}j+a_{3}kig|~i^{2}=j^{2}=k^{2}=ijk=-1,a_{0}%,a_{1},a_{2},a_{3}inmathbb{R}ig}.$ $A=(a_{ij})inmathbb{H}^{m imes n}$,$A^{st}=(ar{a}_{ji})inmathbb{H}^{n imes m}$, $A^{(st)}=(ar{a}_{m-j+1,n-i+1})inmathbb{H}^{n imes m}$, where $ar{a}_{ji}$ is theconjugate of the quaternion $a_{ji}$. We call that $Ainmathbb{H}^{n imesn}$ is $eta$-Hermitian if $A=-eta A^{st}eta$, $etain{i,j,k}$;$Ainmathbb{H}^{n imes n}$ is $eta$-bihermitian if $A=-eta A^{st}%eta=-eta A^{(st)}eta$. We in this paper consider a system of linearreal quaternion matrix equations involving $eta$-Hermicity, i.e.egin{align}%&A_{1}X=C_{1},~XB_{1}=D_{1},
onumber\&A_{2}Y=C_{2},~YB_{2}=D_{2},
onumber\&C_{3}XC_{3}^{etast}+D_{3}YD_{3}^{etast}=A_{3}. label{01111}%end{align}We present some necessary andsufficient conditions for the existence of the $eta$-Hermitian solution tothe system of linear real quaternion matrix equations (
ef{01111}) and give an expression of the $eta$-Hermitian solution to system(
ef{01111}) when it is solvable. As an application, we consider thenecessary and sufficient conditions for the systemegin{equation}A_{b}X=C_{b},~XB_{b}=D_{b},~C_{c}XC_{c}^{etast}=A_{3} label{02222}%end{equation}to have an $eta$-bihermitian solution. We establish an expression of the$eta$-bihermitian to system (
ef{02222}) when it is solvable. We alsoobtain a criterion for a quaternion matrix to be $eta$-bihermitian. Moreover,we provide an algorithm and a numerical example to illustrate the theorydeveloped in this paper. |
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| Keywords:System ofquaternion matrix equations; η-Hermitian solution; $eta$-bihermitiansolution; Quaternion involutions; Moore-Penrose inverse |
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