What form is the complex Jordan form allowed to take?

Question

Could you please explain to me why is the last matrix not included in $(5.32)$?

How is the last matrix different form the three matrices in $(5.32)$.

Answer 1

Because the matrices$$\begin{bmatrix}\lambda&1&0\\0&\lambda&0\\0&0&\lambda\end{bmatrix}\quad\text{and}\quad\begin{bmatrix}\lambda&0&0\\0&\lambda&1\\0&0&\lambda\end{bmatrix}\tag1$$are similar (the author proved that) and therefore, if we want to have all types of matrices which are of the form$$\begin{bmatrix}\lambda&\varepsilon_1&0\\0&\lambda&\varepsilon_2\\0&0&\lambda\end{bmatrix}$$(with each $\varepsilon_j$ equal to $0$ or to $1$) such that every $3\times3$ matrix whose only eigenvalue is $\lambda$ is similar to one of them, it is redundant to have both matrices $(1)$.

Answer 2

Jordan normal forms are only unique up to reordering of the blocks. Thus, the last matrix is the same Jordan normal form as the second in (5.32); i.e. if one is a Jordan normal form of a given matrix, so is the other.