Verifying $k$-Contraction Without Computing $k$-Compounds

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 computation may be difficult to perform analytically and expensive numerically because the
$k$
-additive compound of an
$n\times n$
matrix has dimensions
$\binom{n}{k}\times \binom{n}{k}$
. This article establishes a duality relation between the
$k$
and
$(n-k)$
compounds of an
$n\times n$
matrix
$A$
. This duality relation is used to derive a sufficient condition for
$k$
-contraction that does not require the computation of any
$k$
-compounds. These theoretical results are demonstrated 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 thus admits 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.