正规矩阵谱变化的截断奇异值界
A Truncated Singular-Value Bound for Spectral Variation of Normal Matrices
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中文总结 AI 辅助
针对正规矩阵谱变化，提出仅用差矩阵前一半奇异值的截断界，改进经典Hoffman-Wielandt估计，并在3≤n≤16时优于无维度界。
中文摘要 AI 辅助
对于正规矩阵A和B，经典的Hoffman-Wielandt定理通过A-B的Frobenius范数来界定它们谱之间的最优匹配距离。我们证明了更精确的估计d(σ(A),σ(B))² ≤ ∑_{k=1}^{⌊(n+1)/2⌋} s_k(A-B)²，其中s_k(M)是M的奇异值，仅涉及差矩阵的前⌊(n+1)/2⌋个奇异值。因此，d(σ(A),σ(B)) ≤ √⌊(n+1)/2⌋‖A-B‖，这改进了对所有3 ≤ n ≤ 16的经典无维度界。证明结合了最优匹配距离的极小极大对偶性、谱子空间重叠以及矩形压缩下奇异值的单调性。
英文摘要
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.
发表机构
- Zhili College, Tsinghua University（清华大学智理书院）
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