The Karcher Mean on $2\times 2$ Triangular Matrices with Positive Diagonal Entries

Let $\operatorname{UT}_n$ denote the group of $n\times n$ real upper triangular matrices with positive diagonal entries. We derive a closed-form analytic expression for the Karcher mean on $\operatorname{UT}_2$ and obtain a sharp convergence threshold for the projection mean iteration with fixed ini