Verifying $k$-Contraction without Computing $k$-Compounds

Abstract

Compound matrices have found applications in many fields of science including systems and control theory. In particular, a sufficient condition for $k$-contraction is that a logarithmic norm (also called matrix measure) of the $k$-additive compound of the Jacobian is uniformly negative. However, this may be difficult to check in practice because the $k$-additive compound of an $n\times n$ matrix has dimensions $\binom{n}{k}\times \binom{n}{k}$. For an $n\times n$ matrix $A$, we prove a duality relation between the $k$ and $(n-k)$ compounds of $A$. We use this duality relation to derive a sufficient condition for $k$-contraction that does not require the computation of any $k$-compounds. We demonstrate our results by deriving a sufficient condition for $k$-contraction of an $n$-dimensional Hopfield network that does not require to compute any compounds. In particular, for $k=2$ this sufficient condition implies that the network is $2$-contracting and this implies a strong asymptotic property: every bounded solution of the network converges to an equilibrium point, that may not be unique. This is relevant, for example, when using the Hopfield network as an associative memory that stores patterns as equilibrium points of the dynamics.