A matrix of Jordan form (also Jordan normal form, Jordan canonical form and classical canonical form) is a block matrix containing Jordan blocks along the diagonal and zeros elsewhere. For example:
|3 1 0 0 0 0 0|
|0 3 1 0 0 0 0|
|0 0 3 1 0 0 0|
|0 0 0 3 0 0 0|
|0 0 0 0 5 0 0|
|0 0 0 0 0 2 1|
|0 0 0 0 0 0 2|
All matrices are similar to a matrix in Jordan form. The process of finding the matrix in Jordan form that is similar to a given matrix is called Jordan matrix decomposition. Such matrices are useful for computional purposes. If you wanted to find the nth power of a matrix A, then you could compute it easily by first finding the matrices S and J such that:
A = SJS-1
An = (SJS-1)n = SJS-1SJS-1SJS-1...SJS-1 = SJIJIJ...S-1 = SJnS-1
Jn can be found by computing the nth power of each of of the individual Jordan blocks. The ability to easily compute large powers of matrices is useful for computing the long term behavior of a system of equations.
Sources:
http://en.wikipedia.org/wiki/Jordan_normal_form
Eric W. Weisstein et al. "Jordan Canonical Form." From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/JordanCanonicalForm.html