On stability of invariant subspaces of commuting matrices

We study the stability of (Joint) invariant subspaces of a finite set of commuting matrices. We generalize some of the results of Gohberg, Lancaster, and Rodman for the single matrix case. For sets of two or more commuting matrices we exhibit some phenomena different from the single matrix case. We show that each root subspace is a stable invariant subspace, that each invariant subspace of a root subspace of a nonderogatory eigenvalue is stable, and that, even in the derogatory case, the eigenspace is stable if it is one-dimensional. We prove that a pair of commuting matrices has only finitely many stable invariant subspaces. At the end, we discuss the stability of invariant subspaces of an algebraic multiparameter eigenvalue problem. (C) 2002 Elsevier Science Inc. All rights reserved.