Linear Algebra, 5th Edition

7.2 The Jordan Canonical Form II

For the purposes of this section, we fix a linear operator T on an n-dimensional vector space V such that the characteristic polynomial of T splits. Let ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_k$ be the distinct eigenvalues of T.

By Theorem 7.7 (p. 484), each generalized eigenspace ${\text{K}}_{{\lambda }_i}$ contains an ordered basis ${\beta }_i$ consisting of a union of disjoint cycles of generalized eigenvectors corresponding to ${\lambda }_i$. So by Theorems 7.4(b) (p. 480) and 7.5 (p. 482), the union $\beta ={\displaystyle \underset{i=1}{\overset{k}{\cup }}{\beta }_i}$ is a Jordan canonical basis for T. For each i, let ${\text{T}}_i$ be the restriction of T to ${\text{K}}_{{\lambda }_i}$, and let $A_i={[{\text{T}}_i]}_{{\beta }_i}$. Then $A_i$ is the Jordan canonical form of ${\text{T}}_i$, and

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