The main concern of this paper is the Karcher mean of linearly independent triples (A, B, C) on the hyperbolic manifold of 2 x 2 positive definite matrices of determinant 1. We show that the Karcher mean is of the form (A, B, C) = xA + y(B + C), 0 < x, y and x + 2y < 1 under the trace condition tr(AB(-1)) = tr(AC(-1)). We further find an invertible hyperbolic matrix M depending only on the trace values tr(AB(-1)) and tr(BC-1) such that [x y] = M [cosh theta sinh theta] for some (unique) theta is an element of R. (c) 2022 Elsevier Inc. All rights reserved.
Positive definite matrixKarcher meanCS-decompositionHyperboloid of two sheetsHyperbolic matrixPositively stable matrix
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