  The Lyapunov exponent characterizes the asymptotic behavior of long matrix
products. Recognizing scenarios where the Lyapunov exponent is strictly
positive is a fundamental challenge that is relevant in many applications. In
this work we establish a novel tool for this task by deriving a quantitative
lower bound on the Lyapunov exponent in terms of a matrix sum which is
efficiently computable in ergodic situations. Our approach combines two deep
results from matrix analysis --- the $n$-matrix extension of the
Golden-Thompson inequality and the Avalanche-Principle. We apply these bounds
to the Lyapunov exponents of Schr\"odinger cocycles with certain ergodic
potentials of polymer type and arbitrary correlation structure. We also derive
related quantitative stability results for the Lyapunov exponent near aligned
diagonal matrices and a bound for almost-commuting matrices.
