On Simultaneous Block-Diagonalization of Cyclic Commuting Matrices

We study simultaneous block-diagonalization of cyclic d-tuples of commuting matrices. Some application to ideal projectors are also presented. In particular we extend Hans Stetter’s theorem characterizing Lagrange projectors. 1 Introduction Let V be a …nite-dimensional space over complex …eld C and let L := (L1; :::; Ld) be a d-tuple of pairwise commuting operators on V . Every polynomial p(x1; :::; xd) = X ck1;:::kdx k1 1 :::x kd d 2 C[x1; :::; xd] de…nes an operator p(L) := X ck1;:::kdL k1 1 :::L kd d (1.1) on V . A d-tuple L is called cyclic if there exists a vector v0 2 V such that fp(L)v0; p 2 C[x1; :::; xd]g = V . (1.2) A vector v0 satisfying (1.2) is called a cyclic vector for L. A vector v 2 V is called a common eigenvector for L if for all j = 1; :::; d there exist j 2 C such that Ljv = jv. The d-tuple = ( 1; :::; d) 2 C is called an eigentuple of L. The set of all eigentuples of L is denoted by (L). In case d = 1, an operator L is cyclic if and only if L is 1-regular, i.e., every eigenspace of L is at most one-dimensional. For d > 1 this is false in both directions as the following example (already used in [4] for di¤erent purposes) demonstrates: