Generalization of the Multiplicative and Additive Compounds of Square Matrices and Contraction Theory in the Hausdorff Dimension

The k multiplicative and k additive compounds of a matrix play an important role in geometry, multilinear algebra, the asymptotic analysis of nonlinear dynamical systems, and in bounding the Hausdorff dimension of fractal sets. These compounds are defined for the integer values of k. Here, we introduce generalizations called the α multiplicative and α additive compounds of a square matrix, with α real. We study the properties of these new compounds and demonstrate an application in the context of the Douady and Oesterlé theorem. Our results lead to a generalization of contracting systems to α-contracting systems, with α real. Roughly speaking, the dynamics of such systems contracts any set with the Hausdorff dimension larger than α. For α =1, they reduce to standard contracting systems. We demonstrate our theoretical results by designing a state-feedback controller for a classical chaotic system, guaranteeing the well-ordered behavior of the closed-loop system.