The Similarity Problem

Abstract

The last two chapters of this book have made extensive use of the Jordan normal form. In working through the examples and exercises it must have become apparent that there is no simple method for finding the Jordan form. By contrast, finding the diagonal form of a symmetric matrix reduces to factoring the characteristic polynomial.¹ Why is this? Why is the Jordan form so much more difficult to find than the diagonal or row echelon form? The reason is that linear algebra alone is not sufficient to solve this problem: One needs other methods and techniques, in particular not just algebra. It is the purpose of this chapter to make this more precise, and to provide a proof in the case of 2 x 2 matrices that no such simple method for finding the Jordan form can exist.