Invariant tolerance relations on positive definite matrices (Q2020656)

The author considers the Cartan-Hadamard Riemannian manifold of \(N\times N\) positive definite Hermitian matrices whose tolerance classes are determined by a closed linear form of their Riemannian geodesics and introduce several invariant tolerance relations. It is shown for one of such relations that it admits a linearly independent ordered pair only when the matrix size is even, and the relation on determinant one matrices is characterized by the geometric mean formula \(A\ast B = \frac{A+B}{\sqrt[N]{\det(A+B)}}\).