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.