Jordan Decomposition

The process of decomposing a square matrix $A$ into this form:

Here:

• $J$ is the Jordan form of $A$
• $P$ is an invertible matrix where the columns are generalized eigenvectors of $A$

Generalizes diagonalization when matrix is not diagonalizable. Every square matrix $A$ is Jordan-decomposable.

Jordan Block

Denoted by $B(\lambda,m)$. An upper triangular matrix $B_{m\times m}$ is a Jordan block iff:

• All $m$ diagonal elements are the same eigenvalue ($\lambda$) and
• All super-diagonal elements are $1$

Jordan Form

$J$ from the decomposition.

Here:

• $\lambda_i$ is an eigenvalue of $A$ for $i=1,2,\dots,m$
• $m_i$ is the algebraic multiplicity of $\lambda_i$

Matrix of Generalized Vectors

$P$ from the decomposition.