Numerical Range and Robustness of Grassmannian Frames
Year: 2026
Type: article
Abstract: Grassmannian frames provide redundant vector representations with low mutual coherence, making them relevant to signal recovery, communication, and erasure-resilient reconstruction. This paper develops an operator-theoretic and geometric framework for analysing finite-dimensional Grassmannian frames through numerical range theory and complex projective geometry. The numerical ranges of the frame operator and Gramian operator are characterised in terms of their spectral extrema. For a frame with eigenvalues bounded by \(\lambda\)min and \(\lambda\)max, the frame operator has numerical range [\(\lambda\)min, \(\lambda\)max], while the Gramian operator has numerical range [0, \(\lambda\)max] when the frame is redundant. Tightness is shown to be equivalent to the degeneration of the frame-operator numerical range to a single point. The Rayleigh map on complex projective space is further related to the numerical range, with its critical points corresponding to eigenvectors of the frame operator. Explicit low-dimensional examples illustrate tight and non-tight configurations and their numerical ranges. Erasure resilience is analysed using masking matrices and associated error operators, leading to lower bounds for the remaining frame operator after multiple erasures. For one erasure, equal-norm Parseval frames are characterised as the optimal Parseval configurations, with worst-case error equal to m/M. The results provide a unified description of spectral geometry, tightness, and erasure robustness for finite-dimensional Grassmannian frame systems.
Authors: Arvince Ogendi, Jared Ongaro, Stephen Luketero
Institution: University of Nairobi
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
OpenAlex ID: W7208720786