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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?

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2 Answers 2

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Not sure if it will answer your specific query, but here is a book you can take a look at anyway: Deza, Michel Marie and Deza, Elena: Encyclopedia of distances, (Springer, Heidelberg)

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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.

(This is not exactly your case, but perhaps you can adapt some of the arguments there).

Note however, that even after calculating the geodesics, the actual calculation of the distance can still be intractable.

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  • $\begingroup$ I only could calculate geodesics on very special cases. Couriously I known a formula for the distance between two matrices that appears in the book of Small I cited in the question. If you know something about Voroni diagrams in riemaniann manifolds, it can help to me. $\endgroup$
    – Hideyuki Kabayakawa
    Commented Apr 30, 2016 at 12:37
  • $\begingroup$ 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. $\endgroup$
    – Asaf Shachar
    Commented Apr 30, 2016 at 12:59
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    $\begingroup$ 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). $\endgroup$
    – Asaf Shachar
    Commented Apr 30, 2016 at 13:02

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