TITLE: References for metrics in matrix groups
QUESTION [5 upvotes]: 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?

REPLY [3 votes]: 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.