Jordan canonical form

Abstract

This chapter is of a distinctly more difficult nature than the preceding ones. In it we will treat the problem of finding a particular canonical, simple, matrix, representative of a linear transformation. In the preceding chapter we treated this problem for self-adjoint linear transformations in a finite-dimensional inner product space. For a self-adjoint linear transformation

$$ {\rm{T}}:V \to V $$

in the finite-dimensional inner product space V we saw that we could always find an orthonormal basis for V so that the corresponding matrix of T was diagonal. The required basis would be composed of the normed eigenvectors of T and the diagonal entries of the corresponding matrix are the eigenvalues of T. The problem we now consider is: Can we achieve a similar normal form in general? An answer to this question leads to the Jordan normal form. Before we arrive at this goal we will require quite a few preparatory steps.