Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Scaled Relative Graph of Normal Matrices

Xinmeng Huang, Ernest K. Ryu, Wotao Yin

TL;DR

The paper advances the theory of the Scaled Relative Graph (SRG) by fully characterizing the SRG for linear operators, focusing on block-diagonal and normal matrices within a hyperbolic geometry framework. It defines the SRG via $z_A(x)=\frac{\|Ax\|}{\|x\|}\exp[i\angle(Ax,x)]$ and shows that block-diagonal SRGs decompose as unions of arc-edge polygons ${\mathrm{Poly}}(z_1,...,z_m)$. For normal matrices, it proves ${\mathcal{G}}^+(A)= {\mathrm{Poly}}(\Lambda(A)\cap \mathbb{C}^+)$, establishing invariance under orthogonal similarity and providing explicit 2×2 and symmetric-case descriptions (e.g., ${\mathcal{G}}(A)=\mathrm{Disk}(\lambda_1,\lambda_m)\setminus\bigcup_{i=1}^{m-1}\mathrm{Disk}^\circ(\lambda_i,\lambda_{i+1})$). The results offer a geometric, hyperbolic-theory lens for understanding convergence of fixed-point iterations via the SRG and relate the SRG to the eigenvalue spectrum in a precise, structured way.

Abstract

The Scaled Relative Graph (SRG) is a geometric tool that maps the action of a multi-valued nonlinear operator onto the 2D plane, used to analyze the convergence of a wide range of iterative methods. As the SRG includes the spectrum for linear operators, we can view the SRG as a generalization of the spectrum to multi-valued nonlinear operators. In this work, we further study the SRG of linear operators and characterize the SRG of block-diagonal and normal matrices.

Scaled Relative Graph of Normal Matrices

TL;DR

The paper advances the theory of the Scaled Relative Graph (SRG) by fully characterizing the SRG for linear operators, focusing on block-diagonal and normal matrices within a hyperbolic geometry framework. It defines the SRG via and shows that block-diagonal SRGs decompose as unions of arc-edge polygons . For normal matrices, it proves , establishing invariance under orthogonal similarity and providing explicit 2×2 and symmetric-case descriptions (e.g., ). The results offer a geometric, hyperbolic-theory lens for understanding convergence of fixed-point iterations via the SRG and relate the SRG to the eigenvalue spectrum in a precise, structured way.

Abstract

The Scaled Relative Graph (SRG) is a geometric tool that maps the action of a multi-valued nonlinear operator onto the 2D plane, used to analyze the convergence of a wide range of iterative methods. As the SRG includes the spectrum for linear operators, we can view the SRG as a generalization of the spectrum to multi-valued nonlinear operators. In this work, we further study the SRG of linear operators and characterize the SRG of block-diagonal and normal matrices.

Paper Structure

This paper contains 4 sections, 6 theorems, 48 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Arc-edge polygon and arc-convexity
  3. SRGs of block-diagonal matrices
  4. SRGs of normal matrices

Key Result

Lemma 2.1

\newlabellem:arc-convex20 Let $z_1,\dots,z_m\in\mathbb{C}^+$ and $m\ge 1$. Then $\mathrm{Poly}(z_1,\dots,z_m)$ is "convex" in the following non-Euclidean sense: If $\mathrm{Poly}(z_1,\dots,z_m)$ has an interior, then there is $\{\zeta_1,\dots,\zeta_q\}\subseteq\{z_1,\dots,z_m\}$ such that is a Jordan curve, and the region the curve encloses is $\mathrm{Poly}(z_1,\dots,z_m)$.

Figures (3)

  • Figure 1: Illustration of $\mathrm{Circ}(z_1,z_2)$ and $\mathrm{Arc}_\mathrm{min}(z_1,z_2)$.
  • Figure 1: Illustration of Proposition \ref{['mat-2*2']}
  • Figure 2: Illustration of Theorem \ref{['mat-main-thm2']}. For normal matrices, multiplicity of eigenvalues do not affect the SRG.

Theorems & Definitions (11)

  • Lemma 2.1
  • Proof 1
  • Theorem 3.1
  • Proof 2
  • Proposition 4.1
  • Proof 3
  • Proposition 4.2
  • Proof 4
  • Theorem 4.3
  • Proof 5
  • ...and 1 more