Matrix nearness problems: Do real inputs admit real solutions?
Abstract
Given a real square matrix A and a nonempty closed target set of complex matrices ℰ, invariant by complex conjugation and containing real matrices, does the subset of ℰ consisting of the matrices nearest to A in the Frobenius distance always contain a real matrix? We give negative answers when 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≥ 3, and this holds for every distance induced by a unitarily invariant norm. For both problems and n ≥ 3, we show that a real input A uniformly drawn from the unit sphere |A|F=1 has no real minimizer over ℰ with probability strictly between $0$ and $1$. Beyond these two cases, we prove for an arbitrary nonempty closed target sets ℰ that, for almost every real input, either ℰ contains a unique nearest matrix which is real, or every nearest matrix in ℰ is nonreal.
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BibTeX
@misc{noferini2026matrix,
title = {{Matrix nearness problems: Do real inputs admit real solutions?}},
author = {Vanni Noferini and Matvei Zhukov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.14828},
eprint = {2609.14828},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14828},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Matrix nearness problems: Do real inputs admit real solutions?
AU - Vanni Noferini
AU - Matvei Zhukov
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2609.14828
UR - https://arxiv.org/abs/2609.14828
AB - Given a real square matrix A and a nonempty closed target set of complex matrices ℰ, invariant by complex conjugation and containing real matrices, does the subset of ℰ consisting of the matrices nearest to A in the Frobenius distance always contain a real matrix? We give negative answers when 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≥ 3, and this holds for every distance induced by a unitarily invariant norm. For both problems and n ≥ 3, we show that a real input A uniformly drawn from the unit sphere |A|_F=1 has no real minimizer over ℰ with probability strictly between $0$ and $1$. Beyond these two cases, we prove for an arbitrary nonempty closed target sets ℰ that, for almost every real input, either ℰ contains a unique nearest matrix which is real, or every nearest matrix in ℰ is nonreal.
ER -