In linear algebra, a nilpotent matrix is a square matrix N such that for some positive integer k. The smallest such k is called the index of N, sometimes the degree of N. More generally, a nilpotent transformation is a linear transformation L of a vector space such that L^k = 0 for some positive integer k (and thus, L^j = 0 for all j \geq k). Both of these concepts are special cases of a more general concept of nilpotence that applies to elements of rings.

Examples

Example 1

The matrix is nilpotent with index 2, since A^2 = 0.

Example 2

More generally, any n-dimensional triangular matrix with zeros along the main diagonal is nilpotent, with index \le n. For example, the matrix is nilpotent, with The index of B is therefore 4.

Example 3

Although the examples above have a large number of zero entries, a typical nilpotent matrix does not. For example, although the matrix has no zero entries.

Example 4

Additionally, any matrices of the form such as or square to zero.

Example 5

Perhaps some of the most striking examples of nilpotent matrices are n\times n square matrices of the form: The first few of which are: These matrices are nilpotent but there are no zero entries in any powers of them less than the index.

Example 6

Consider the linear space of polynomials of a bounded degree. The derivative operator is a linear map. We know that applying the derivative to a polynomial decreases its degree by one, so when applying it iteratively, we will eventually obtain zero. Therefore, on such a space, the derivative is representable by a nilpotent matrix.

Characterization

For an n \times n square matrix N with real (or complex) entries, the following are equivalent: The last theorem holds true for matrices over any field of characteristic 0 or sufficiently large characteristic. (cf. Newton's identities) This theorem has several consequences, including: See also: Jordan–Chevalley decomposition.

Classification

Consider the n \times n (upper) shift matrix: This matrix has 1s along the superdiagonal and 0s everywhere else. As a linear 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 n, and is the canonical nilpotent matrix. Specifically, if N is any nilpotent matrix, then N 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 N 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 L on naturally determines a flag of subspaces and a signature The signature characterizes L 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 T is locally nilpotent if for every vector v, there exists a such that For operators on a finite-dimensional vector space, local nilpotence is equivalent to nilpotence.