Spectral scales and linear pencils

Developed in 1999 by Akemann, Anderson, and Weaver, the spectral scale of an $n\times n$ matrix $A$, is a convex, compact subset of ${\mathbb{R}}^3$ that reveals important spectral information about $A$ \cite{AAW}. In this paper we present new information found in the spectral scale of a matrix. Given a matrix $A=A_1+i{A}_2$ with $A_1$ and $A_2$ self-adjoint and $A_2\mathrm{\ne }0,$ we show that faces in the boundary of the spectral scale of $A$ that are parallel to the x-axis describe elements of $\sigma (A_1,A_2)\bigcap \mathbb{R},$ the real elements of the spectrum of the linear pencil $P(\lambda )=A_1+\lambda A_2\mathrm{.}$