 Jordan normal form

Over a field in which the characteristic polynomial splits, every finite-dimensional linear operator has a matrix that is a direct sum of Jordan blocksThe block sizes for each eigenvalue are unique up to ordering. A block of size greater than one records the failure of diagonalizability and consists of a Jordan chain of generalized eigenvectors.^[ref]

For $J_2(\lambda )=\lambda I+N$ with $N^2=0$, its exponential is $e^{t{J}_2(\lambda )}=e^{\lambda t}(I+tN)$. Thus nontrivial blocks produce polynomial factors as well as exponential growth or decay in a linear differential equation. Over $\mathbb{R}$, a matrix with nonreal eigenvalues requires complexification for this form.

Jordan normal form (or Jordan canonical form) is a special form of a square matrix in linear algebra that simplifies the representation of linear transformations. It is particularly useful for studying the properties of linear operators and can be used to perform calculations related to matrix exponentiation, differential equations, and more. A matrix is said to be in Jordan normal form if it is a block diagonal matrix composed of Jordan blocks.