Newton-Type Methods For Simultaneous Matrix Diagonalization

TL;DR

This paper introduces a Newton-type algorithm for simultaneously diagonalizing commuting matrices, achieving quadratic convergence without solving linear systems, supported by theoretical analysis and numerical experiments.

Contribution

It presents a novel Newton-type method for simultaneous matrix diagonalization that converges quadratically and does not rely on solving linear systems.

Findings

Quadratic convergence under certain conditions

Effective numerical performance demonstrated

Theoretical analysis supports convergence claims

Abstract

This paper proposes a Newton-type method to solve numerically the eigenproblem of several diagonalizable matrices, which pairwise commute. A classical result states that these matrices are simultaneously diagonalizable. From a suitable system of equations associated to this problem, we construct a sequence that converges quadratically towards the solution. This construction is not based on the resolution of a linear system as is the case in the classical Newton method. Moreover, we provide a theoretical analysis of this construction and exhibit a condition to get a quadratic convergence. We also propose numerical experiments, which illustrate the theoretical results.