Nilpotent matrix

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

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

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 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 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 despite the fact that none of the matrices prior to the matrix turning into zero contain any zero entries.[5]

Characterization

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 degree 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.

Classification

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 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 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

  • If is nilpotent, then is invertible, where is the identity matrix. The inverse is given by
where only finitely many terms of this sum 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 .

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. http://people.math.sfu.ca/~idmercer/nilpotent.pdf
  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

  • Beauregard, Raymond A.; Fraleigh, John B. (1973), A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields, Boston: Houghton Mifflin Co., ISBN 0-395-14017-X
  • Herstein, I. N. (1975), Topics In Algebra (2nd ed.), John Wiley & Sons
  • Nering, Evar D. (1970), Linear Algebra and Matrix Theory (2nd ed.), New York: Wiley, LCCN 76091646
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