The Diagonalizable and Nilpotent Parts of a Matrix

It is an easy consequence of the Jordan canonical form that a matrix A ∈Mn×n(C) can be decomposed into a sum A = DA + NA where DA is a diagonalizable matrix, NA a nilpotent matrix, and such that DANA = NADA. It is clear that both DA and NA also commute with A. This decomposition is often referred to as the Jordan decomposition and has found many applications throughout the years. For example, it is not hard to see that computing powers of A is much simpler if we know its Jordan decomposition. (The reader may refer to [1] for some of the recent applications of this decomposition.) It is “often” the case that if a matrix B commutes with a given matrix A, then B is in fact a polynomial in A. This is indeed the case for the DA and NA described above. A proof of this fact can be found in [3, §6.8]. The proof presented therein, which perhaps can be best described as “semi constructive” as it invokes a fact about polynomials that depends on the Euclidean algorithm, relies on the decomposition of C into subspaces that are invariant under the operator induced by the matrix A. The recent work done on this problem is much more technical and has focused on finding fast algorithms to express DA and NA as polynomials in A (e.g., [1, 4]). This article is devoted to an introduction and a new, computer-algebra-system motivated, very elementary solution (acessible to an undergraduate with an upperdivision background in linear algebra) to the problem of expressing DA and NA as polynomials in A. Specifically, we 1.) introduce the diagonalizable + nilpotent decomposition; 2.) present a simple (perhaps the simplest so far as it relies almost exclusively on the Jordan form and matrix algebra), constructive proof of the fact that DA and NA can be expressed as polynomials in A; 3.) discuss some of the consequences our proof; and 4.) state a few questions that arise from our construction. We begin by stating the theorem whose proof will be our focus for the majority of the article. Theorem 1. Let A ∈ Mn×n(C) and let A = DA + NA be a decomposition of A where DA is diagonalizable, NA is nilpotent and NADA = DANA. Then there exists a polynomials p(x), q(x) ∈ P (C) such that p(A) = NA and q(A) = DA. Moreover, NA and DA are unique.