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

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Example 1

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The matrix

 

is nilpotent with index 2, since  .

Example 2

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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 4.

Example 3

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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

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Additionally, any matrices of the form

 

such as

 

or

 

square to zero.

Example 5

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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

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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

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For an   square matrix   with real (or complex) entries, the following are equivalent:

  •   is nilpotent.
  • The characteristic polynomial for   is  .
  • The minimal polynomial for   is   for some positive integer  .
  • The only complex eigenvalue for   is 0.

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:

  • The index of an   nilpotent matrix is always less than or equal to  . For example, every   nilpotent matrix squares to zero.
  • The determinant and trace of a nilpotent matrix are always zero. Consequently, a nilpotent matrix cannot be invertible.
  • The only nilpotent diagonalizable matrix is the zero matrix.

See also: Jordan–Chevalley decomposition#Nilpotency criterion.

Classification

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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

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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

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  • If   is nilpotent of index   , then   and   are invertible, where   is the   identity matrix. The inverses are given by
     
  • If   is nilpotent, then
     

    Conversely, if   is a matrix and

     
    for all values of  , then   is nilpotent. In fact, since   is a polynomial of degree  , it suffices to have this hold for   distinct values of  .
  • Every singular matrix can be written as a product of nilpotent matrices.[8]
  • A nilpotent matrix is a special case of a convergent matrix.

Generalizations

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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

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  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

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