The sharp spectral-matching constant for normal matrices
For normal $A, B \in \mathbb{C}^{n \times n}$ with eigenvalues $\alpha_1, \dots, \alpha_n$ and $\beta_1, \dots, \beta_n$, each listed with algebraic multiplicity, let $$d_\infty(A, B) = \min_{\pi \in S_n} \max_{1 \le j \le n} |\alpha_j - \beta_{\pi(j)}|$$ be the optimal matching distance between their spectra, let $\|\cdot\|_2$ be the spectral norm, and let $$C_{\mathrm{normal}} = \sup_{n \ge 2}\ \sup_{\substack{A, B \in \mathbb{C}^{n \times n} \text{ normal} \\ A \ne B}} \frac{d_\infty(A, B)}{\|A - B\|_2},$$ the least $C$ with $d_\infty(A, B) \le C\|A - B\|_2$ for all normal matrices $A$ and $B$ of the same order. If $A$ and $B$ are both Hermitian, or both unitary, this inequality holds with $C = 1$. Bhatia, Davis and McIntosh proved that $C_{\mathrm{normal}}$ is finite, and Holbrook that $C_{\mathrm{normal}} > 1$. A $3 \times 3$ pair of normal matrices due to Krause has $d_\infty(A, B)^2 = 28/13$ and $\|A - B\|_2^2 = 27/13$, so $C_{\mathrm{normal}} \ge \sqrt{28/27}$. It is known that $C_{\mathrm{normal}} < 3$.
Problem. Determine $C_{\mathrm{normal}}$.
Source: Bhatia (2007), Perturbation Bounds for Matrix Eigenvalues, Chapter 9, pp. 154–155; restated by Zhan (2008), Problem 27.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- NA — Numerical Analysis
- Secondary
- SP
- Keywords
- normal matrices, OpenProblemsInNLA, perturbation bounds, spectral variation
- Literature checked
- Oct. 7, 2026 by clemens-admin
References
@book{bhatia2007,
author = {Bhatia, Rajendra},
title = {Perturbation Bounds for Matrix Eigenvalues},
series = {Classics in Applied Mathematics},
volume = {53},
publisher = {SIAM},
address = {Philadelphia},
year = {2007},
doi = {10.1137/1.9780898719079},
note = {Reprint of the 1987 original with supplements; Chapter 9, Spectral variation of normal matrices, pp. 152--160},
}
@incollection{zhan2008,
author = {Zhan, Xingzhi},
title = {Open problems in matrix theory},
booktitle = {Proceedings of the Fourth International Congress of Chinese Mathematicians, Vol. {I}},
editor = {Ji, Lizhen and Liu, Kefeng and Yang, Lo and Yau, Shing-Tung},
publisher = {Higher Education Press},
address = {Beijing},
year = {2008},
pages = {367--382},
}
@article{bhatia1983,
author = {Bhatia, Rajendra and Davis, Chandler and McIntosh, Alan},
title = {Perturbation of spectral subspaces and solution of linear operator equations},
journal = {Linear Algebra Appl.},
volume = {52/53},
year = {1983},
pages = {45--67},
doi = {10.1016/0024-3795(83)80007-X},
}
@article{holbrook1992,
author = {Holbrook, John A.},
title = {Spectral variation of normal matrices},
journal = {Linear Algebra Appl.},
volume = {174},
year = {1992},
pages = {131--144},
doi = {10.1016/0024-3795(92)90047-E},
}
@misc{semrl2009,
author = {{\v{S}}emrl, Peter},
title = {Geometrical techniques in linear algebra},
howpublished = {Lecture notes, Summer School and Advanced Workshop on Trends and Developments in Linear Algebra, ICTP, Trieste},
year = {2009},
url = {https://indico.ictp.it/event/a08167/session/73/contribution/52/material/0/0.pdf},
note = {Section 3.2, Theorems 5 and 6},
}
@misc{openproblemsinnla-sp07,
author = {Townsend, Alex},
title = {{SP-07}: The sharp spectral-matching constant for normal matrices},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/eigenvalues-and-inverse-problems/SP-07/README.md},
note = {Read at commit 7336302, 2026-10-06},
}
Resolution claims
Exact rational pairs of normal matrices are constructed whose ratio $d_\infty(A,B)/\|A-B\|_2$ exceeds the classical value $\sqrt{28/27}$ of Krause's example. The strongest has dimension seven and proves $C_{\mathrm{normal}}>103077/100000$. Its proof consists of an exact orthogonal-reflection construction, a deficient matching block, and an exact positive-definite Gram certificate. The arithmetic is checked by two separate implementations, one using only Python's standard library and …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-09 | Open → Partially resolved | The registry's curators | Stronger lower bounds and dimension reductions v1 recorded from OpenProblemsInNLA @ eb7c9d6, SP-07/report.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00155, "The sharp spectral-matching constant for normal matrices", clopen.solutions.