Linear Algebra with Applications, 10th Edition

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 $Y\prime =\mathit{\text{AY}}$, 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 ${\lambda }_1,\dots ,{\lambda }_k$, then V can be decomposed into a direct sum of invariant subspaces $S_1,\dots ,S_k$ such that $L-{\lambda }_i⌶$ is nilpotent on $S_i$ for each $i=1,\dots ,k$. 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 ...