Every square matrix is similar to one in Jordan form.

Theorem: Every square matrix over $\mathbb{C}$ is similar to a matrix in Jordan canonical form.

That is, for any $A \in \mathbb{C}^{n \times n}$, there exists an invertible matrix $P$ such that:

$$P^{-1}AP = J$$

where $J$ is a block diagonal matrix composed of Jordan blocks:

$$J = \begin{pmatrix} J_1 & & \\ & \ddots & \\ & & J_k \end{pmatrix}$$

Each Jordan block $J_i$ corresponds to one eigenvalue $\lambda$ and has $\lambda$ on the diagonal, 1 on the superdiagonal, and 0 elsewhere.

Significance: The Jordan form is the closest any matrix can get to being diagonal. A matrix is diagonalizable if and only if all Jordan blocks have size 1.

The Jordan form is unique up to the ordering of the blocks.