8.2 The Jordan Canonical Form

In this section, we will show that any linear operator L on an n-dimensional vector space V can be represented by a block diagonal matrix whose diagonal blocks are simple Jordan matrices. We will apply this result to solving systems of linear differential equations of the form Y=AY, where A is defective.

Let us begin by considering the case where L has more than one distinct eigenvalue. We wish to show that if L has distinct eigenvalues λ1,,λk, then V can be decomposed into a direct sum of invariant subspaces S1,,Sk such that Lλi is nilpotent on Si for each i=1,,k. To do this, we must first prove the following lemma and theorem.

Lemma 8.2.1

If L is a linear operator mapping an n-dimensional vector space ...

Get Linear Algebra with Applications, 10th Edition now with the O’Reilly learning platform.

O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.