Fast Singular Value Decay for Lyapunov Solutions with Nonnormal Coefficients

Lyapunov equations with low-rank right-hand sides often have solutions whose singular values decay rapidly, enabling iterative methods that produce low-rank approximate solutions. All previously known bounds on this decay involve quantities that depend quadratically on the departure of the coefficient matrix from normality: these bounds suggest that the larger the departure from normality, the slower the singular values will decay. We show this is only true up to a threshold, beyond which a larger departure from normality can actually correspond to faster decay of singular values: if the singular values decay slowly, the numerical range cannot extend far into the right half-plane.