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

Let ${\mathrm{UT}}_n$ 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 ${\mathrm{UT}}_2$ and obtain a sharp convergence threshold for the projection mean iteration with fixed initial diagonal, together with a sufficient convergence condition for arbitrary initial points. In addition, we obtain explicit formulas for the log-Euclidean mean and establish a Lie–Trotter type formula for the Karcher mean on ${\mathrm{UT}}_2$. For ${\mathrm{UT}}_n$ we derive the diagonal and first superdiagonal entries of every solution of the Karcher equation and transfer nonconvergence examples from ${\mathrm{UT}}_2$.