A real nearest matrix with spectrum in the closed left half-plane
For $\mathbb{F} \in \{\mathbb{R}, \mathbb{C}\}$ and $n \ge 1$ let $\mathcal{H}_n(\mathbb{F})$ be the set of matrices $B \in \mathbb{F}^{n \times n}$ all of whose eigenvalues $\lambda$ satisfy $\operatorname{Re} \lambda \le 0$: the closed left half-plane, with no condition on the Jordan structure of eigenvalues on the imaginary axis. For $A \in \mathbb{R}^{n \times n}$ let $$d_{\mathbb{F}}(A) = \min_{B \in \mathcal{H}_n(\mathbb{F})} \|A - B\|_F,$$ where $\|\cdot\|_F$ is the Frobenius norm; the minimum exists because $\mathcal{H}_n(\mathbb{F})$ is closed and nonempty. Clearly $d_{\mathbb{C}}(A) \le d_{\mathbb{R}}(A)$. Noferini and Poloni found, in all the examples they computed, a real minimizer that also solves the complex problem. The set $\mathcal{H}_n(\mathbb{C})$ is not convex, and the real part of a matrix in it need not lie in $\mathcal{H}_n(\mathbb{R})$. They proved that every real minimizer has a complex Schur decomposition that is a stationary point of their Riemannian reformulation of the complex problem.
Problem. Is $d_{\mathbb{R}}(A) = d_{\mathbb{C}}(A)$ for every $n \ge 1$ and every real matrix $A \in \mathbb{R}^{n \times n}$? Equivalently, does the complex minimization problem always have a real minimizer?
Source: Noferini and Poloni (2021), Section 7.6, Conjecture 1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- NA — Numerical Analysis
- Secondary
- OC
- Keywords
- AIM Open Applied Problems, Hurwitz stable matrices, matrix nearness problems
References
@article{noferini2021,
author = {Noferini, Vanni and Poloni, Federico},
title = {Nearest {$\Omega$}-stable matrix via {R}iemannian optimization},
journal = {Numer. Math.},
volume = {148},
number = {4},
year = {2021},
pages = {817--851},
doi = {10.1007/s00211-021-01217-4},
eprint = {2002.07052},
archivePrefix = {arXiv},
}
@misc{noferini2026zhukov,
author = {Noferini, Vanni and Zhukov, Matvei},
title = {Matrix nearness problems: Do real inputs admit real solutions?},
year = {2026},
eprint = {2609.14828},
archivePrefix = {arXiv},
}
@misc{aim-nearest-hurwitz-real-minimizer,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A real nearest matrix with spectrum in the closed left half-plane}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/578-nearest-hurwitz-real-minimizer.md},
note = {Entry 578 at commit a153f46, 2026-10-13},
}
Resolution claims
Given a real square matrix $A$ and a nonempty closed target set of complex matrices $\mathcal E$, invariant by complex conjugation and containing real matrices, does the subset of $\mathcal{E}$ consisting of the matrices nearest to $A$ in the Frobenius distance always contain a real matrix? We give negative answers when $\mathcal{E}$ is either the set of normal matrices (solving …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-15 | Needs verification → Solution claimed | — | Recorded here: arXiv:2609.14828v1. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00320, "A real nearest matrix with spectrum in the closed left half-plane", clopen.solutions.