# Nearest Omega-stable matrix via Riemannian optimization

arXiv: https://arxiv.org/abs/2002.07052 · HTML: https://mlsanity.com/paper/2002.07052/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Feb 2020.

## Built on by (most cited first, 1 of 1)

- [A Novel Schur-Decomposition-Based Weight Projection Method for Stable State-Space Neural-Network Architectures](https://mlsanity.com/paper/2605.14489/) (2026)

## Abstract (arXiv)

We study the problem of finding the nearest Omega-stable matrix to a certain matrix A, i.e., the nearest matrix with all its eigenvalues in a prescribed closed set Omega. Distances are measured in the Frobenius norm. An important special case is finding the nearest Hurwitz or Schur stable matrix, which has applications in systems theory. We describe a reformulation of the task as an optimization problem on the Riemannian manifold of orthogonal (or unitary) matrices. The problem can then be solved using standard methods from the theory of Riemannian optimization. The resulting algorithm is remarkably fast on small-scale and medium-scale matrices, and returns directly a Schur factorization of the minimizer, sidestepping the numerical difficulties associated with eigenvalues with high multiplicity.