# References for metrics in matrix groups

I am studying a very concrete matrix group with a riemaniann (right invariant) metric for solving a question on Applied Math. I need explicit formulas for the distance between two matrices, geodesics and so on.

The matrix group I am studying is $UT(n)$, the $n\times n$ upper triangular matrices $\Pi=(\Pi_{i j})$ with positive diagonal elements and $\Pi_{11}=1$; but specially $UT(3)$.

The metric is defined in the book of C.G. Small The Statistical Theory of Shape ( p. 101-106): Let $\Pi_x$ and $\Pi_{x+dx}$ in $UT(n)$ and let $\Lambda = \Pi_{x+dx}\cdot \Pi_{x}^{-1}$. The infinitesimal distance ds from $\Pi_x$ to $\Pi_{x+dx}$ is given by the formula $$ds^2= \frac{\sum (\lambda_j-\bar\lambda)^2}{n}$$ where $\lambda_j$ are the eigenvalues of $\Lambda^T\Lambda$ and $\bar\lambda=\frac {\sum \lambda_j}{n}$.

Looking for Lie groups I found references for metric questions, but in a very general setting of differential geometry.

In the other hand, I found introduction references for matrix groups, but they avoid metric considerations.

And thus my question: Are there references in matrix groups that avoid generalities but have a lot of results in the metric structure?

This paper might give you some ideas on how to calculate the geodesics. It is about left invariant metrics on $GL_n(\mathbb{R})$. The geodesics are calculated using their characterization as critical points of the energy functional.
• Unfortunately I am not familiar with Voroni diagrams in the Riemannian context. At the risk of self-publicizing, I might suggest another approach for the calculation of the geodesics: For invariant metrics on a Lie group, there is a technique which reduces the calculation of the geodesics, to a differential equation in a single vector space,$T_eG$. The idea is to choose an orthonormal basis for $(T_eG,g_e)$, and push it forward to a global $g$-orthonormal frame on $G$. Then you need to compute the $1$-forms of the Levi-Civita connection w.r.t this frame. – Asaf Shachar Apr 30 '16 at 12:59
• Finally, using the invariance of the metric you can hopefully reduce everything to an equation in $T_eG$. (Essentially because the $1$-forms will be constant functions). A detailed treatment of this approach can be found in my paper here: arxiv.org/abs/1603.05868 (In the appendix, after proposition A.1). – Asaf Shachar Apr 30 '16 at 13:02