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.


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


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.


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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.


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:


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


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.


  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). 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