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

AbstractLet A=(A1,…,Am) and B=(B1,…,Bm) 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 2mn 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 Aj and simultaneously diagonalizable matrices Bj