Metraipolis › archive › dc_e098fbde2ff3

Matrix nearness problems: Do real inputs admit real solutions? — arXiv:2609.14828

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.14828
License (as archived): CC0-1.0
Archived: 2026-10-16T00:39:30.841Z
SHA-256 of the archived text: c2a4ee89e80e9c87ab401538ea32f092b88432add94f9aae93b2002b11936382
Stable URL of this page: https://metraipolis.com/documento/dc_e098fbde2ff3

How to cite

Matrix nearness problems: Do real inputs admit real solutions? — arXiv:2609.14828 — arXiv (en), https://arxiv.org/abs/2609.14828 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_e098fbde2ff3 (Metraipolis archive dc_e098fbde2ff3, sha256 c2a4ee89e80e9c87ab401538ea32f092b88432add94f9aae93b2002b11936382)

BibTeX

@misc{dce098fbde2ff3,
  title = {Matrix nearness problems: Do real inputs admit real solutions? — arXiv:2609.14828},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.14828},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_e098fbde2ff3; sha256 c2a4ee89e80e9c87ab401538ea32f092b88432add94f9aae93b2002b11936382},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.14828. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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 a question posed by N. Higham) or the set of matrices whose eigenvalues lie in the closed left half-plane (solving a question posed by the first author and F. Poloni). For the latter problem, we also argue that the ratio between the real and complex distances is unbounded in every dimension $n\geq 3$, and this holds for every distance induced by a unitarily invariant norm. For both problems and $n \geq 3$, we show that a real input $A$ uniformly drawn from the unit sphere $\|A\|_F=1$ has no real minimizer over $\mathcal{E}$ with probability strictly between $0$ and $1$. The paper is complemented by some further results that are valid for more general target sets $\mathcal{E}$.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-15 in math.OC).
Categorie arXiv: Numerical Analysis (math.NA) ; Optimization and Control (math.OC).
Pagina del lavoro: https://arxiv.org/abs/2609.14828 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-16T00:16:48.869Z dalla pagina https://arxiv.org/catchup/math.OC/2026-09-15?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).