  Data which lie in the space $\mathcal{P}_{m\,}$, of $m \times m$ symmetric
positive definite matrices, (sometimes called tensor data), play a fundamental
role in applications including medical imaging, computer vision, and radar
signal processing. An open challenge, for these applications, is to find a
class of probability distributions, which is able to capture the statistical
properties of data in $\mathcal{P}_{m\,}$, as they arise in real-world
situations. The present paper meets this challenge by introducing Riemannian
Gaussian distributions on $\mathcal{P}_{m\,}$. Distributions of this kind were
first considered by Pennec in $2006$. However, the present paper gives an exact
expression of their probability density function for the first time in existing
literature. This leads to two original contributions. First, a detailed study
of statistical inference for Riemannian Gaussian distributions, uncovering the
connection between maximum likelihood estimation and the concept of Riemannian
centre of mass, widely used in applications. Second, the derivation and
implementation of an expectation-maximisation algorithm, for the estimation of
mixtures of Riemannian Gaussian distributions. The paper applies this new
algorithm, to the classification of data in $\mathcal{P}_{m\,}$, (concretely,
to the problem of texture classification, in computer vision), showing that it
yields significantly better performance, in comparison to recent approaches.
