Invariant tolerance relations on positive definite matrices

In this paper we introduce several invariant tolerance relations on the Cartan-Hadamard Riemannian manifold of N×N positive definite Hermitian matrices whose tolerance classes are determined by a closed linear form of their Riemannian geodesics. 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#B=A+Bdet⁡(A+B)N. Weighted and multivariate versions of the relation are discussed.