Spectral scales and linear pencils

2005/11/04 by Christopher M. Pavone, Pavone, Christopher M.
Computer Science · Mathematics · #15A22 (primary) #47A10 (secondary) #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Matrix Theory and Algorithms #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.FA #math.SP #msc:15A22 #msc:47A10

paper · pdf · doi:10.48550/arxiv.math/0511120

6 pages, 3 figures

openalex publication_date 2005/11/04 · arxiv created 2005/11/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Developed in 1999 by Akemann, Anderson, and Weaver, the spectral scale of an n× n matrix A, is a convex, compact subset of ℝ³ that reveals important spectral information about A \citeAAW. In this paper we present new information found in the spectral scale of a matrix. Given a matrix A=A₁ + iA₂ with A₁ and A₂ self-adjoint and A₂≠ 0, we show that faces in the boundary of the spectral scale of A that are parallel to the x-axis describe elements of σ(A₁,A₂)\bigcapℝ, the real elements of the spectrum of the linear pencil P(λ)=A₁ + λA₂.

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