Section 30.1 Motivation
We continue our quest to answer Question 28.1.1.
In Chapter 28 and Chapter 29, we took two different paths from the initial case of diagonalizable matrices. In the former we generalized from diagonal to roughly diagonal in blocks, and in the latter we moved from diagonal form to triangular form, albeit a particularly simple example of it. In this chapter, we merge the two branches back together.
Philosophy of inquiry: boldly forge ahead using whatever tools worked before.
Apply the tools we have already developed in previous cases to see what works and what doesn't, with the hope that we'll be able to adjust whatever doesn't work.
Generalized eigenvectors were central to handling the case of scalar-triangular form, where we assumed each matrix had a single repeated eigenvalue. Can we still use generalized eigenvectors to achieve a triangular form for matrices with more than one eigenvalue? In that case we will be dealing with several different generalized eigenspaces. Block-diagonal form gives us a way to break the action of \(A\) on \(\R^n\) (or \(\C^n\)) into its action on each in a collection of smaller subspaces. We know that eigenspaces are always \(A\)-invariant (Proposition 28.6.3) and form an independent set (Theorem 28.6.10). Will generalized eigenspaces do the same? Let's find out.