THE DIAGONALIZABLE AND NILPOTENT PARTS OF A MATRIX

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 articl