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A Truncated Singular-Value Bound for Spectral Variation of Normal Matrices
Abstract
For normal matrices A and B, the classical Hoffman-Wielandt theorem bounds the optimal matching distance between their spectra by the Frobenius norm of A-B. We prove the sharper estimate d(σ(A),σ(B))^2 \le \sum_{k=1}^{\lfloor (n+1)/2 \rfloor} s_k(A-B)^2, where s_k(M) are the singular values of M, involving only the first \lfloor (n+1)/2 \rfloor singular values of the difference. Consequently, d(σ(A),σ(B)) \le \sqrt{\lfloor (n+1)/2 \rfloor}\,\|A-B\|, which improves the classical dimension-free bound for all 3 \le n \le 16. The proof combines a min-max duality for optimal matching distances with spectral subspace overlaps and the monotonicity of singular values under rectangular compressions.
Related
- Spectral density estimation for normal matrices
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BibTeX
@misc{tang2026truncated,
title = {{A Truncated Singular-Value Bound for Spectral Variation of Normal Matrices}},
author = {Qiyue Tang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.09177},
eprint = {2609.09177},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09177},
}
RIS
TY - UNPB
TI - A Truncated Singular-Value Bound for Spectral Variation of Normal Matrices
AU - Qiyue Tang
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2609.09177
UR - https://arxiv.org/abs/2609.09177
AB - For normal matrices A and B, the classical Hoffman-Wielandt theorem bounds the optimal matching distance between their spectra by the Frobenius norm of A-B. We prove the sharper estimate d(σ(A),σ(B))^2 \le \sum_{k=1}^{\lfloor (n+1)/2 \rfloor} s_k(A-B)^2, where s_k(M) are the singular values of M, involving only the first \lfloor (n+1)/2 \rfloor singular values of the difference. Consequently, d(σ(A),σ(B)) \le \sqrt{\lfloor (n+1)/2 \rfloor}\,\|A-B\|, which improves the classical dimension-free bound for all 3 \le n \le 16. The proof combines a min-max duality for optimal matching distances with spectral subspace overlaps and the monotonicity of singular values under rectangular compressions.
ER -