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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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