Definition of Jordan form

Definition: The Jordan canonical form (or Jordan normal form) of a square matrix is a block diagonal matrix composed of Jordan blocks:

$$J = \begin{pmatrix} J_1 & 0 & \cdots & 0 \\ 0 & J_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & J_k \end{pmatrix}$$

Each Jordan block $J_i$ has the form:

$$J_i = \begin{pmatrix} \lambda & 1 & 0 & \cdots & 0 \\ 0 & \lambda & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda & 1 \\ 0 & 0 & \cdots & 0 & \lambda \end{pmatrix}$$

Theorem: Every square matrix over $\mathbb{C}$ is similar to a matrix in Jordan form. The form is unique up to reordering of blocks.

• The number of Jordan blocks for eigenvalue $\lambda$ equals the geometric multiplicity of $\lambda$
• The sum of sizes of Jordan blocks for $\lambda$ equals the algebraic multiplicity
• A matrix is diagonalizable iff all Jordan blocks have size 1