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PreprintarXiv licencearXiv

On a Conjecture Related to Eigenvalue Perturbations

  • Lei-Hong Zhang
  • Ren-Cang Li
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Abstract

In connection with eigenvalue perturbation bounds, Li [Linear Algebra Appl., 278 (1998), pp. 317-326] formulated three minimization problems involving Hadamard products. The third asks how to relate minW nonsingular|W∘ G|ui |W−1∘ GT|ui and minP permutation|P∘ G|ui2, where |·|ui denotes a unitarily invariant norm and ∘ denotes the Hadamard product, i.e., entrywise multiplication. The first quantity never exceeds the second, and a reverse inequality with a constant independent of the matrix size was conjectured. Romeo and Tilli [Linear Algebra Appl., 326 (2001), pp. 161-172] showed that if such a bound holds for every G, then |·|ui must be uniformly equivalent to the Frobenius norm |·|F. We prove that the two quantities coincide when |·|ui=|·|F: for every complex matrix G, minW nonsingular |W∘ G|F |W−1∘ GT|F =minP permutation|P∘ G|F2. The associated conjecture therefore holds with the optimal constant c=1.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026conjecture,
  title = {{On a Conjecture Related to Eigenvalue Perturbations}},
  author = {Lei-Hong Zhang and Ren-Cang Li},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.25552},
  eprint = {2609.25552},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.25552},
}
RIS
TY  - UNPB
TI  - On a Conjecture Related to Eigenvalue Perturbations
AU  - Lei-Hong Zhang
AU  - Ren-Cang Li
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.25552
UR  - https://arxiv.org/abs/2609.25552
AB  - In connection with eigenvalue perturbation bounds, Li [Linear Algebra Appl., 278 (1998), pp. 317-326] formulated three minimization problems involving Hadamard products. The third asks how to relate min_W nonsingular|W∘ G|_ui |W^−1∘ G^T|_ui and min_P permutation|P∘ G|_ui^2, where |·|_ui denotes a unitarily invariant norm and ∘ denotes the Hadamard product, i.e., entrywise multiplication. The first quantity never exceeds the second, and a reverse inequality with a constant independent of the matrix size was conjectured. Romeo and Tilli [Linear Algebra Appl., 326 (2001), pp. 161-172] showed that if such a bound holds for every G, then |·|_ui must be uniformly equivalent to the Frobenius norm |·|_F. We prove that the two quantities coincide when |·|_ui=|·|_F: for every complex matrix G, min_W nonsingular |W∘ G|_F |W^−1∘ G^T|_F =min_P permutation|P∘ G|_F^2. The associated conjecture therefore holds with the optimal constant c=1.
ER  -