## A real quaternion matrix equation with applications.(English)Zbl 1317.15016

Let $$\mathbb{H}^{m\times n}$$ be the set of all $$m\times n$$ matrices over the real quaternion algebra $\mathbb{H}=\{a_0+a_1i+a_2j+a_3k|\,i^2=j^2=k^2=ijk=-1,\, a_0,a_1,a_2,a_3\in\mathbb{R}\}.$ For $$A\in\mathbb{H}^{m\times n}$$, it is denoted that $$A^{\eta}=-\eta A\eta$$, and $$A^{\eta^*}=-\eta A^*\eta$$, where $$\eta\in\{i,j,k\}$$, and $$A^*$$ is the conjugate transpose of $$A$$, and the map $$A\mapsto A^{\eta^*}$$ is an involution. A matrix $$A\in\mathbb{H}^{n\times n}$$ is called $$\eta$$-Hermitian if $$A^{\eta^*}=A$$ for $$\eta\in\{i,j,k\}$$.
In the paper, the real quaternion matrix equation $A_1X+(A_1X)^{\eta^*}+B_1YB_1^{\eta^*}+C_1ZC_1^{\eta^*}=D_1$ is considered. For the case when $$D_1$$ is $$\eta$$-Hermitian, necessary and sufficient conditions on matrices $$A_1$$, $$B_1$$, $$C_1$$, and $$D_1$$ are established for the equation above to be solvable with respect to the triplet $$(X,Y,Z)$$, where $$Y$$ and $$Z$$ are required to be $$\eta$$-Hermitian. The explicit solution is presented and the minimal ranks of the solutions $$Y$$ and $$Z$$ are found.

### MSC:

 15A24 Matrix equations and identities 15A09 Theory of matrix inversion and generalized inverses 15B57 Hermitian, skew-Hermitian, and related matrices 11R52 Quaternion and other division algebras: arithmetic, zeta functions 15B33 Matrices over special rings (quaternions, finite fields, etc.)
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