  The $k$ multiplicative and $k$ additive compounds of a matrix play an
important role in geometry, multi-linear algebra, the asymptotic analysis of
nonlinear dynamical systems, and in bounding the Hausdorff dimension of fractal
sets. These compounds are defined for integer values of $k$. Here, we introduce
generalizations called the $\alpha$ multiplicative and $\alpha$ additive
compounds of a square matrix, with $\alpha$ real. We study the properties of
these new compounds and demonstrate an application in the context of the Douady
and Oesterl\'{e} Theorem. This leads to a generalization of contracting systems
to $\alpha$ contracting systems, with $\alpha$ real. Roughly speaking, the
dynamics of such systems contracts any set with Hausdorff dimension larger than
$\alpha$. For $\alpha=1$ they reduce to standard contracting systems.
