Lyapunov Exponent of Rank One Matrices: Ergodic Formula and Inapproximability of the Optimal Distribution

Abstract

The Lyapunov exponent corresponding to a set of square matrices A = {A 1 , ... , A n } and a probability distribution p over {1, ... , n} is λ(A, p) := lim k→∞ 1/k E log ||A σk ⋯ A σ2 A σ1 ||, where σ i are i.i.d. according to p. This quantity is of fundamental importance to control theory since it determines the asymptotic convergence rate e λ(A,p) of the stochastic linear dynamical system x k+1 = A σk x k . This paper investigates the following “design problem”: given A, compute the distribution p minimizing λ(A, p). Our main result is that it is NP-hard to decide whether there exists a distribution p for which λ(A, p) <; 0, i.e. it is NP-hard to decide whether this dynamical system can be stabilized. This hardness result holds even in the “simple” case where A contains only rank-one matrices. Somewhat surprisingly, this is in stark contrast to the Joint Spectral Radius - the deterministic kindred of the Lyapunov exponent - for which the analogous optimization problem over switching rules is known to be exactly computable in polynomial time for rank-one matrices. To prove this hardness result, we first observe that the Lyapunov exponent of rank-one matrices admits a simple formula and in fact is a quadratic form in p. Hardness of the design problem is shown through a reduction from the Independent Set problem. Along the way, simple examples are given illustrating that p → λ(A, p) is neither convex nor concave in general, and a connection is made to the fact that the Martin distance on the (1, n) Grassmannian is not a metric.