Nilpotent matrix

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.

ExamplesEdit

Example 1Edit

The matrix

 

is nilpotent with index 2, since  .

Example 2Edit

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

 

is nilpotent, with

 

The index of   is therefore 4.

Example 3Edit

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 4Edit

Additionally, any matrices of the form

 

such as

 

or

 

square to zero.

Example 5Edit

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]

CharacterizationEdit

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.
  • tr(Nk) = 0 for all  .

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.

ClassificationEdit

Consider the   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 subspacesEdit

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 propertiesEdit

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

As long as   is nilpotent, both sums converge, as only finitely many terms are nonzero.

  • If   is nilpotent, then
 
where   denotes the   identity matrix. 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  .

GeneralizationsEdit

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.

NotesEdit

  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). math.sfu.ca. self-published; personal credentials: PhD Mathematics, Simon Fraser University. Retrieved 22 August 2020.
  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

ReferencesEdit

External linksEdit