The Scaled Relative Graph of a Linear Operator

Abstract: The scaled relative graph (SRG) of an operator is a subset of the complex plane. It captures several salient features of an operator, such as contractiveness, and can be used to reveal the geometric nature of many of the inequality based arguments used in the convergence analyses of fixed point iterations. In this paper we show that the SRG of a linear operator can be determined from the numerical range of a closely related linear operator. Furthermore we demonstrate that the SRG of a linear operator has a range of spectral and convexity properties, and satisfies an analogue of Hildebrant's theorem.