Spectral Scales and Linear Pencils

Developed in 1999 by Akemann, Anderson, and Weaver, the spectral scale of an n × n matrix A, is a convex, compact subset of R that reveals important spectral information about A [4]. 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 6= 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) ⋂ R, the real elements of the spectrum of the linear pencil P (λ) = A1 + λA2. 1 Preliminary Definitions Notation. Let M = Mn(C), the set of n×n matrices with complex entries, and let < ·, · > denote the standard inner-product on C. For each C ∈ M, we let τ(C) = 1 n tr(C), where tr is the standard canonical trace on M. Given A ∈ M, A will denote the adjoint of A, σ(A) will denote the set of eigenvalues (spectrum) of A, and W (A) will denote the numerical range of A. We let ||A|| denote the operator (spectral) norm of A. Definition 1.1. Given A ∈ M, write A = A1 + iA2 with A1 and A2 selfadjoint. The spectral scale of A is defined as B(A) = { (τ(C), τ(A1C), τ(A2C)) : C ∈ M + 1 } ⊂ R, where M+1 denotes the positive semidefinite part of the unit ball of M.