Year: 2026
Type: preprint
Abstract: Spectral projectors and the reflectors derived from them are basic objects in numerical linear algebra. This paper studies the prescribed-index reflector I-2P_k, where P_k is the spectral projector associated with the first k eigenvectors of a symmetric matrix. If a shift s lies in the target spectral gap, then this reflector is exactly the shifted matrix sign sign(H-sI). The exact identity is elementary, but its algorithmic consequences are not: all admissible shifts give the same exact reflector, while finite-step sign filters can have very different errors. We analyze odd sign-preserving spectral filters, prove local inheritance and discrete stability for the induced reflector iterations, derive a gap-dependent Newton-Schulz operator bound, and give deterministic admissibility certificates for inexact and reused shifts. The analysis identifies the shifted spectral margin as the quantity controlling finite-step reflector accuracy and explains why the midpoint shift is the natural default. Numerical experiments separate the matrix-function issues from the outer saddle-search dynamics: controlled spectra verify the margin predictions, low-dimensional tests distinguish shifted signs from raw signs, target-index scans probe non-small k, and Allen-Cahn and dense timing tests identify the regimes in which full-matrix sign filters are useful and the stiff regimes in which stronger sign engines are needed.
Source: arXiv (Cornell University)
Author: Zhao, Jin
Institution: Capital Normal University
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7162443467
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