Notes

Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrixthe normal form for a given M is not entirely unique An n × n matrix A is diagonalizable if and only if the sum of the dimensions of the eigenspaces is n
Given an eigenvalue λi, its geometric multiplicity is the dimension of Ker(A − λi I), and it is the number of Jordan blocks corresponding to λi.[10]The sum of the sizes of all Jordan blocks corresponding to an eigenvalue λi is its algebraic multiplicity.[10]
A is diagonalizable if and only if, for every eigenvalue λ of A, its geometric and algebraic multiplicities coincide.
The Jordan block corresponding to λ is of the form λ I + N, where N is a nilpotent matrix defined as Nij = δi,j−1 (where δ is the Kronecker delta). The nilpotency of N can be exploited when calculating f(A) where f is a complex analytic function. For example, in principle the Jordan form could give a closed-form expression for the exponential exp(A)
Given an eigenvalue λi, its multiplicity in the minimal polynomial is the size of its largest Jordan block.
Therefore, the statement that every square matrix A can be put in Jordan normal form is equivalent to the claim that there exists a basis consisting only of eigenvectors and generalized eigenvectors of A.
Let λ1, ..., λq be the distinct eigenvalues of A, and si be the size of the largest Jordan block corresponding to λi. It is clear from the Jordan normal form that the minimal polynomial of A has degree Σsi.

While the Jordan normal form determines the minimal polynomial, the converse is not true. This leads to the notion of elementary divisors. The elementary divisors of a square matrix A are the characteristic polynomials of its Jordan blocks. The factors of the minimal polynomial m are the elementary divisors of the largest degree corresponding to distinct eigenvalues.