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

@article{Altschuler2020LyapunovEO, title={Lyapunov Exponent of Rank-One Matrices: Ergodic Formula and Inapproximability of the Optimal Distribution}, author={Jason M. Altschuler and Pablo A. Parrilo}, journal={SIAM J. Control. Optim.}, year={2020}, volume={58}, pages={510-528} }

The Lyapunov exponent corresponding to a set of square matrices $\mathcal{A} = \{A_1, \dots, A_n \}$ and a probability distribution $p$ over $\{1, \dots, n\}$ is $\lambda(\mathcal{A}, p) := \lim_{k...

## 2 Citations

### Matrix Concentration for Products

- MathematicsFoundations of Computational Mathematics
- 2021

This paper develops nonasymptotic growth and concentration bounds for a product of independent random matrices. These results sharpen and generalize recent work of Henriksen–Ward, and they are…

### Approximating Min-Mean-Cycle for low-diameter graphs in near-optimal time and memory

- Computer Science, MathematicsSIAM Journal on Optimization
- 2022

This work revisits Min-Mean-Cycle and gives a much faster approximation algorithm that, for graphs with polylogarithmic diameter, has near-linear runtime and is the first algorithm whose runtime for the complete graph scales in the number of vertices as $\tilde{O}(n^2)$.

## References

SHOWING 1-10 OF 46 REFERENCES

### The Lyapunov exponent and joint spectral radius of pairs of matrices are hard—when not impossible—to compute and to approximate

- MathematicsMath. Control. Signals Syst.
- 1997

It is shown that the joint and generalized spectral radii of two integer matrices are not approximable in polynomial time, and that two related quantities—the lower spectral radius and the largest Lyapunov exponent—are not algorithmically approxIMable.

### Lower bounds for the maximal Lyapunov exponent

- Mathematics
- 1990

Upper bounds for the maximal Lyapunov exponent,E, of a sequence of matrix-valued random variables are easy to come by asE is the infimum of a real-valued sequence. We shall show that under…

### Zeta function for the Lyapunov exponent of a product of random matrices.

- Mathematics, PhysicsPhysical review letters
- 1992

A cycle expansion for the Lyapunov exponent of a product of random matrices is derived by using a Bernoulli dynamical system to mimic the randomness.

### An upper bound for the largest Lyapunov exponent of a Markovian product of nonnegative matrices

- MathematicsTheor. Comput. Sci.
- 2005

### Convex optimization methods for computing the Lyapunov exponent of matrices

- Computer Science, Mathematics2013 European Control Conference (ECC)
- 2013

A new approach is introduced to evaluate the largest Lyapunov exponent of a family of matrices, which describes the stability with probability one of a randomly switching linear system, and a new universal upper bound is derived and a similar lower bound does not exist.

### Joint Spectral Radius and Path-Complete Graph Lyapunov Functions

- Mathematics, Computer ScienceSIAM J. Control. Optim.
- 2014

This work defines a class of graphs called path-complete graphs, and shows that any such graph gives rise to a method for proving stability of the switched system, which enables several asymptotically tight hierarchies of semidefinite programming relaxations that unify and generalize many existing techniques.

### A note on common quadratic Lyapunov functions for linear inclusions: Exact results and Open Problems

- MathematicsProceedings of the 44th IEEE Conference on Decision and Control
- 2005

We prove several exact results on approximability of joint spectral radius by matrix norms induced by Euclidean norms. We point out, perhaps for the first time in this context, a difference between…

### Joint Spectral Characteristics of Matrices: A Conic Programming Approach

- Mathematics, Computer ScienceSIAM J. Matrix Anal. Appl.
- 2010

A new method to compute the joint spectral radius and the Joint spectral subradius of a set of matrices by a lifting procedure and the efficiency of this algorithm is demonstrated by applying it to several problems in combinatorics, number theory, and discrete mathematics.