In linear algebra, a nilpotent matrix is a square matrix N such that

for some positive integer . The smallest such is called the index of ,[1] sometimes the degree of .

More generally, a nilpotent transformation is a linear transformation of a vector space such that for some positive integer (and thus, for all ).[2][3][4] 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 .

Example 2

More generally, any -dimensional triangular matrix with zeros along the main diagonal is nilpotent, with index [citation needed]. For example, the matrix

is nilpotent, with


The index of is therefore 3.

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 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.[5]

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

This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (May 2018) (Learn how and when to remove this template message)

For an square matrix 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#Nilpotency criterion.

Classification

Consider the (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:

[6]

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.[7]

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 b1b2 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

  1. ^ Herstein (1975, p. 294)
  2. ^ Beauregard & Fraleigh (1973, p. 312)
  3. ^ Herstein (1975, p. 268)
  4. ^ Nering (1970, p. 274)
  5. ^ Mercer, Idris D. (31 October 2005). "Finding "nonobvious" nilpotent matrices" (PDF). idmercer.com. self-published; personal credentials: PhD Mathematics, Simon Fraser University. Retrieved 5 April 2023.
  6. ^ Beauregard & Fraleigh (1973, p. 312)
  7. ^ Beauregard & Fraleigh (1973, pp. 312, 313)
  8. ^ R. Sullivan, Products of nilpotent matrices, Linear and Multilinear Algebra, Vol. 56, No. 3

References