In linear algebra, the Jordan normal form answers the question, for a given square matrix M over a field K, to what extent M can be simplified into a standard shape by changing basis. It is not possible to make all such M diagonal, even when K is algebraically closed: what the Jordan normal form does is to quantify the failure. In abstract terms, any M is written as a sum M' + M* where M' is diagonalizable, M* is nilpotent, and M' commutes with M*.
The way the normal form is usually stated writes out explicitly what that implies about M as a sum of block square matrices along the leading diagonal (with zero blocks elsewhere). The typical such Jordan block is cI + N, where N is the special nilpotent matrix with (i,j)th entry 1 if i = j+1, and otherwise 0 (acts on basis vector ek by decrementing k by 1). This form is valid over an algebraically closed field.
The proof of the Jordan normal form is usually carried out as an application to the ring K[X] of the structure theorem for finitely-generated modules over principal ideal domains, of which it is a corollary.