The decomposition of idempotents associated with inflators

Abstract

In a recent paper, Friedland, Hershkowitz, and Schneider introduced a new matrix product called the inflation product and a new class of matrices called inflators. Fundamental to the constructions were certain idempotent matrices associated with the inflators. This paper studies the structure of the idempotent matrix associated with an inflator. In particular, it is shown that if the idempotent associated with an inflator has rank greater than one, then the idempotent can be split into several pairwise orthogonal idempotents of lower rank such that the resultant idempotents are associated with inflators which are inflation product factors of the original inflator. The indecomposable idempotents associated with the decomposition of an inflator are characterized in terms of rank, and are shown to be generally nonunique. The number of indecomposable idempotents in a splitting is shown to be invariant.