Theoretical analysis establishes criteria for decomposing division ring matrices into positive semidefinite products, extending Ballantine's decomposition theorem.
In this paper, we investigate the Dieudonné determinant and the definition of a positive semidefinite matrix over a division ring. We establish necessary and sufficient conditions for the decomposition of matrices into a product of a finite number of positive semidefinite matrices over division rings and one diagonalizable matrix, whose Dieudonné determinant is the commutator class containing one. Consequently, it is possible to immediately obtain Ballantine's decomposition over the real quaternion division ring and complex number field.
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Gatephan et al. (2026) studied this question.
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