  We consider the joint asymptotic distribution of \emph{best} and
$\varepsilon$ Diophantine approximates of matrices in several aspects. Our main
results describe the resulting limiting measures for almost every matrix.
Multiplicative Diophantine approximation is treated for the first time in this
context and a number of Diophantine corollaries are derived including a matrix
L\'{e}vy-Khintchine type theorem. While we treat the general case of
approximation of matrices, our results are already new for the case of
simultaneous Diophantine approximation of vectors. Our approach is dynamical
and is inspired by work of several authors, notably Shapira and Weiss. It is
based on the construction of an appropriate Poincar\'{e} section for certain
diagonal group actions on the space of unimodular lattices. The main new idea
in our paper is a method which allows us to treat actions of higher rank
groups.

Number Theory;Dynamical Systems