# Definition of Jordan form

Put content here.**Definition:** The **Jordan canonical form** (or Jordan normal form) of a square matrix is a block diagonal matrix composed of **Jordan blocks**:
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$$J = \begin{pmatrix} J_1 & 0 & \cdots & 0 \\ 0 & J_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & J_k \end{pmatrix}$$
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Each Jordan block $J_i$ has the form:
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$$J_i = \begin{pmatrix} \lambda & 1 & 0 & \cdots & 0 \\ 0 & \lambda & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda & 1 \\ 0 & 0 & \cdots & 0 & \lambda \end{pmatrix}$$
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**Theorem:** Every square matrix over $\mathbb{C}$ is similar to a matrix in Jordan form. The form is unique up to reordering of blocks.
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- 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

# Parents

* Canonical forms of matrices