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Real representation approach to quaternion matrix equation involving \(\varphi \)-Hermicity. (English) Zbl 1435.15022

Summary: For a quaternion matrix \(A\), we denote by \(A_\phi\) the matrix obtained by applying \(\varphi\) entrywise to the transposed matrix \(A^T\), where \(\varphi\) is a nonstandard involution of quaternions. \(A\) is said to be \(\varphi \)-Hermitian or \(\varphi \)-skew-Hermitian if \(A = A_\phi\) or \(A = - A_\phi \), respectively. In this paper, we give a complete characterization of the nonstandard involutions \(\varphi\) of quaternions and their conjugacy properties; then we establish a new real representation of a quaternion matrix. Based on this, we derive some necessary and sufficient conditions for the existence of a \(\varphi \)-Hermitian solution or \(\varphi \)-skew-Hermitian solution to the quaternion matrix equation \(A X = B\). Moreover, we give solutions of the quaternion equation when it is solvable.

MSC:

15B33 Matrices over special rings (quaternions, finite fields, etc.)
15A24 Matrix equations and identities
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References:

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