Reduction to Triangular Form

Abstract

The Primary Decomposition Theorem shows that for a linear mapping f on a finite-dimensional vector space V there is a basis of V with respect to which f can be represented by a block diagonal matrix. As we have seen, in the special situation where the minimum polynomial of f is a product of distinct linear factors, this matrix is diagonal. We now turn our attention to a slightly more general situation, namely that in which the minimum polynomial of f factorises as a product of linear factors that are not necessarily distinct, i.e. is of the form $$ {m_f} = \mathop \Pi \limits_{i = 1}^k {(X - {\lambda _i})^{ei}} $$ where each e i ≥ 1. This, of course, is always the case when the ground field is ℂ, so the results we shall establish will be valid for all linear mappings on a finite-dimensional complex vector space. To let the cat out of the bag, our specific objective is to show that when the minimum polynomial of f factorises completely there is a basis of V with respect to which the matrix of f is triangular. We recall that a matrix A = [aij] n×n is (upper) triangular if a ij = 0 whenever i > j.