A Bauer-Fike theorem for the joint spectrum of commuting matrices

Abstract

Let A=( A 1,…, A m ) and B=( B 1,…, B m ) be m-tuples commuting n by n self-adjoint matrices. We obtain a number ε=‖ Cliff( A− B)‖ such that within a distance ε of each joint eigenvalue of A there is a joint eigenvalue of B. The Clifford operator Cliff( A− B) of A− B can be represented by a square matrix of size 2 m n and is defined using Clifford algebras. When m = 1, ‖ Cliff( A− B)‖ = ‖ A− B‖, the operator bound norm of A - B. Similar results are obtained for arbitrary commuting matrices A j and simultaneously diagonalizable matrices B j .