Jordan block

Definition

Let $F$ be a field, let $\lambda \in F$, and let $k \ge 1$. The Jordan block $J_k(\lambda)$ is the $k \times k$ matrix with entries $$(J_k(\lambda))_{ij} = \begin{cases} \lambda & \text{if } j = i, \\ 1 & \text{if } j = i + 1, \\ 0 & \text{otherwise}. \end{cases}$$

Remarks

For example, $J_3(\lambda) = \begin{pmatrix} \lambda & 1 & 0 \\ 0 & \lambda & 1 \\ 0 & 0 & \lambda \end{pmatrix}$. $J_k(\lambda) - \lambda I_k$ is nilpotent of index $k$. Its only eigenvalue is $\lambda$, with algebraic multiplicity $k$ and geometric multiplicity $1$. Some authors put the $1$ entries below the diagonal.

Explanations

L1 human

A Jordan block is the simplest matrix that is not diagonalisable, when its size is at least $2$. It has one eigenvalue and only one line of eigenvectors. Every complex matrix is built from such blocks, by the Jordan normal form.