# Fast singular value decay for Lyapunov solutions with nonnormal coefficients

arXiv: https://arxiv.org/abs/1410.8741 · HTML: https://mlsanity.com/paper/1410.8741/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Oct 2014.

## Built on by (most cited first, 2 of 2)

- [HOPE for a Robust Parameterization of Long-memory State Space Models](https://mlsanity.com/paper/2405.13975/) (2024)
- [Model Reduction for Nonlinear Systems by Balanced Truncation of State and Gradient Covariance](https://mlsanity.com/paper/2207.14387/) (2022)

## Abstract (arXiv)

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.