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 $A=A_1 + iA_2$ with $A_1$ and $A_2$ self-adjoint and $A_2\neq 0,$ 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 $P(\lambda)=A_1 + \lambda A_2.$