Notation
Some textbooks have the ones on the subdiagonal; that is, immediately below the main diagonal instead of on the superdiagonal. The eigenvalues are still on the main diagonal. See more

If A is a real matrix, its Jordan form can still be non-real. Instead of representing it with complex eigenvalues and ones on the superdiagonal, as discussed above, there exists a real … See more

One can see that the Jordan normal form is essentially a classification result for square matrices, and as such several important results … See more

Iteration of the Jordan chain motivates various extensions to more abstract settings. For finite matrices, one gets matrix functions; this can be extended to compact operators … See more

In general, a square complex matrix A is similar to a block diagonal matrix
$${\displaystyle J={\begin{bmatrix}J_{1}&\;&\;\\\;&\ddots &\;\\\;&\;&J_{p}\end{bmatrix}}}$$
where each block Ji is a square matrix of the form See more

Given an eigenvalue λ, every corresponding Jordan block gives rise to a Jordan chain of linearly independent vectors pi, i = 1, ..., b, … See more

Jordan reduction can be extended to any square matrix M whose entries lie in a field K. The result states that any M can be written as a sum D + N where D is semisimple, N is nilpotent, and DN = ND. This is called the Jordan–Chevalley decomposition See more