On matrix measures and convex Liapunov functions

Abstract

In this paper, we extend the concept of the measure of a matrix to encompass a measure induced by an arbitrary convex positive definite function. It is shown that this "modified" matrix measure has most of the properties of the usual matrix measure, and that many of the known applications of the usual matrix measure can therefore be carried over to the modified matrix measure. These applications include deriving conditions for a mapping to be a diffeomorphism on Rⁿ, and estimating the solution errors that result when a nonlinear network is approximated by a piecewise linear network. We also develop a connection between matrix measures and Liapunov functions. Specifically, we show that if V is a convex positive definite function and A is a Hurwitz matrix, then μ_V(A) < 0, if and only if V is a Liapunov function for the system x=Ax. This linking up between matrix measures and Liapunov functions leads to some results on the existence of a "common" matrix measure μ _V(·) such that μ _V(Ai) < 0 for each of a given set of matrices A₁,..., A_m. Finally, we also give some results for matrices with nonnegative off-diagonal terms.