Almost-commuting matrices are almost jointly diagonalizable

TL;DR

This paper investigates the connection between approximate joint diagonalization of self-adjoint matrices and their commutator norm, demonstrating that matrices that nearly commute can be nearly diagonalized simultaneously.

Contribution

It establishes a quantitative relationship showing that almost commuting self-adjoint matrices are nearly jointly diagonalizable by a unitary transformation.

Findings

Almost commuting matrices are nearly jointly diagonalizable.

The degree of diagonalizability relates to the norm of the commutator.

Provides bounds on how close matrices are to being jointly diagonalizable.

Abstract

We study the relation between approximate joint diagonalization of self-adjoint matrices and the norm of their commutator, and show that almost commuting self-adjoint matrices are almost jointly diagonalizable by a unitary matrix.