On a determinantal inequality arising from diffusion tensor imaging

In comparing geodesics induced by different metrics, Audenaert formulated the following determinantal inequality $\det(A^2+|BA|)\le \det(A^2+AB),$ where $A, B$ are $n\times n$ positive semidefinite matrices. We complement his result by proving $\det(A^2+|AB|)\ge \det(A^2+AB).$ Our proofs feature the fruitful interplay between determinantal inequalities and majorization relations. Some related questions are mentioned.

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