Linear and multilinear algebra; matrix theory Definition

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.

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Matrix definition
Square matrix definition
Jordan block definition

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Jordan normal form

Tag	`0PBFS92AJP`
URI	`m2://0PBFS92AJP`
Kind	definition
Writing	W0 stub
MSC	15A21
Defines	Jordan block
Uses	Square matrix

Jordan block

Definition

Remarks

Reading path

Where this leads

Further reading

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