Jordan Form

A jordan canonical form is a Upper Triangular Matrix version of a given matrix. It appears as:

$$\left[\begin{array}{ccccc}\lambda & 1& 0& \dots & 0\\ 0& \lambda & 1& \ddots & 0\\ ⋮& ⋮& \ddots & \ddots & ⋮\\ 0& 0& 0& \lambda & 1\\ 0& 0& 0& 0& \lambda \end{array}\right]$$

Definition

• Suppose $T\in \mathcal{L}(V)$
• A Jordan Basis $\beta$ allows for jordan matrix:

A_{1} & & 0\ & \ddots & \ 0 & & A_{p} \ \end{array}\right]

- Where, $$A_{i} = \left[\begin{array}{cc} \lambda_{k} & 1 & \dots & 0\\ & \ddots & \ddots & \\ & & \ddots & 1\\ 0 & & & \lambda_{k}\\ \end{array}\right]$$ ### Alternate Definition A jordan form of $A$ exists $\Longleftrightarrow$ the [[Characteristic Polynomial]] of $A$ splits into [[Linear Factor|Linear Factors]] # Guides - [[Finding Jordan Form of Matrix]] # Theorems - [[Jordan Basis Exists for Triangularizable Linear Transformations]] - [[All Operators on A Vector Space Defined Over an Algebraically Closed Field Have A Jordan Form]]