transformation, the shift matrix "shifts" the components of a vector one position to the left, with a zero appearing in the last position: This matrix is nilpotent with degree , and is the canonical nilpotent matrix. Specifically, if is any nilpotent matrix, then is similar to a block diagonal matrix of the form where each of the blocks is a shift matrix (possibly of different sizes). This form is a special case of the Jordan canonical form for matrices. For example, any nonzero 2 × 2 nilpotent matrix is similar to the matrix That is, if is any nonzero 2 × 2 nilpotent matrix, then there exists a basis b1, b2 such that Nb1 = 0 and Nb2 = b1. This classification theorem holds for matrices over any field. (It is not necessary for the field to be algebraically closed.) Flag of subspaces A nilpotent transformation on naturally determines a flag of subspaces and a signature The signature characterizes up to an invertible linear transformation. Furthermore, it satisfies the inequalities Conversely, any sequence of natural numbers satisfying these inequalities is the signature of a nilpotent transformation. Additional properties Generalizations A linear operator is locally nilpotent if for every vector , there exists a such that For operators on a finite-dimensional vector space, local nilpotence is equivalent to nilpotence. Notes References External links Nilpotent matrix and nilpotent transformation on PlanetMath. Matrices