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The reflexive re-nonnegative definite solution to a quaternion matrix equation. (English) Zbl 1147.15012

The authors present first criteria that a \(3\times 3\) partitioned quaternion matrix is re-nonnegative definite and a quaternion matrix is reflexive re-nonnegative definite. The main result of this paper is a necessary and sufficient condition for the existence of re-nonnegative definite solution to the quaternion matrix equation \(A_1 X_1 A_1^* + A_2 X_2 A_2^* = B\) (* stands for conjugate transpose) as well as an expression of the general solution. On this basis a necessary and sufficient condition for the existence and the expression of the reflexive re-nonnegative definite solution to the quaternion matrix equation \(AXA^* = B\).

MSC:

15A24 Matrix equations and identities
15B33 Matrices over special rings (quaternions, finite fields, etc.)
15B57 Hermitian, skew-Hermitian, and related matrices
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