Linear Algebra, 5th Edition

7.1 The Jordan Canonical Form I

Let T be a linear operator on a finite-dimensional vector space V, and suppose that the characteristic polynomial of T splits. Recall from Section 5.2 that the diagonalizability of T depends on whether the union of ordered bases for the distinct eigenspaces of T is an ordered basis for V. So a lack of diagonalizability means that at least one eigenspace of T is too “small.”

In this section, we extend the definition of eigenspace to generalized eigenspace. From these subspaces, we select ordered bases whose union is an ordered basis $\beta$ for V such that

$${[\text{T}]}_{\beta }=\left(\begin{array}{cccc}{A}_1& O& \cdots & O\\ O& A_2& \cdots & O\\ ⋮& ⋮& & ⋮\\ O& O& \cdots & A_k\end{array}\right),$$

where each O is a zero matrix, and each $A_i$