Arithmetic–Geometric Mean determinantal identity

Non-Commutative Sylvester's Determinantal Identity

Sylvester’s identity is a classical determinantal identity with a straightforward linear algebra proof. We present combinatorial proofs of several non-commutative extensions, and find a β-extension that is both a generalization of Sylvester’s identity and the β-extension of the quantum MacMahon master theorem.

Non-commutative Sylvester’s determinantal identity, preprint

Sylvester's identity is a classical determinantal identity with a straightforward linear algebra proof. We present a new, combinatorial proof of the identity, prove several non-commutative versions, and find a β-extension that is both a generalization of Sylvester's identity and the β-extension of the MacMahon master theorem.

Determinantal complexity

A Determinantal Inequality for the Geometric Mean with an Application in Diffusion Tensor Imaging

We prove that for positive semidefinite matrices A and B the eigenvalues of the geometric mean A#B are log-majorised by the eigenvalues of A1/2B1/2. From this follows the determinantal inequality det(I + A#B) ≤ det(I + A1/2B1/2). We then apply this inequality to the study of interpolation methods in diffusion tensor imaging.

A Determinant of the Chudnovskys Generalizing the Elliptic Frobenius-Stickelberger-Cauchy Determinantal Identity

D.V. Chudnovsky and G.V. Chudnovsky [CH] introduced a generalization of the FrobeniusStickelberger determinantal identity involving elliptic functions that generalize the Cauchy determinant. The purpose of this note is to provide a simple essentially non-analytic proof of this evaluation. This method of proof is inspired by D. Zeilberger’s creative application in [Z1]. AMS Subject Classificatio...