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 e_{k} 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.