Nilpotent matrix

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

for some positive integer k. The smallest such k is sometimes called the degree of N.

More generally, a nilpotent transformation is a linear transformation L of a vector space such that Lk = 0 for some positive integer k (and thus, Lj = 0 for all jk). Both of these concepts are special cases of a more general concept of nilpotence that applies to elements of rings.

Examples

The matrix

is nilpotent, since M2 = 0. More generally, any triangular matrix with 0s along the main diagonal is nilpotent. For example, the matrix

is nilpotent, with

Though the examples above have a large number of zero entries, a typical nilpotent matrix does not. For example, the matrix

squares to zero, though the matrix has no zero entries.

Characterization

For an n × n square matrix N 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:

  • The degree of an n × n nilpotent matrix is always less than or equal to n. For example, every 2 × 2 nilpotent matrix squares to zero.
  • The determinant and trace of a nilpotent matrix are always zero.
  • The only nilpotent diagonalizable matrix is the zero matrix.

Classification

Consider the n × n 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 slot to the left:

This matrix is nilpotent with degree n, and is the “canonical” nilpotent matrix.

Specifically, if N is any nilpotent matrix, then N is similar to a block diagonal matrix of the form

where each of the blocks S1S2, ..., Sr is a shift matrix (possibly of different sizes). This theorem is a special case of the Jordan canonical form for matrices.

For example, any nonzero 2 × 2 nilpotent matrix is similar to the matrix

That is, if N 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 L on Rn naturally determines a flag of subspaces

and a signature

The signature characterizes L 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

where only finitely many terms of this sum are nonzero.
  • If N is nilpotent, then
where I denotes the n × n identity matrix. Conversely, if A is a matrix and
for all values of t, then A is nilpotent.

Generalizations

A linear operator T is locally nilpotent if for every vector v, there exists a k such that

For operators on a finite-dimensional vector space, local nilpotence is equivalent to nilpotence.

References

  1. R. Sullivan, Products of nilpotent matrices, Linear and Multilinear Algebra, Vol. 56, No. 3

External links