Nilpotent matrix

Examples

The matrix

M = [ 0 1 0 0 ]

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

N = [ 0 2 1 6 0 0 1 2 0 0 0 3 0 0 0 0 ]

is nilpotent, with

N 2 = [ 0 0 2 7 0 0 0 3 0 0 0 0 0 0 0 0 ] ;   N 3 = [ 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 ] ;   N 4 = [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] .

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

N = [ 5 − 3 2 15 − 9 6 10 − 6 4 ]

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:

N is nilpotent.

The minimal polynomial for N is x^k for some positive integer k ≤ n.

The characteristic polynomial for N is xⁿ.

The only eigenvalue for N is 0.

tr(N^k) = 0 for all k > 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 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. Consequently, a nilpotent matrix cannot be invertible.

The only nilpotent diagonalizable matrix is the zero matrix.

Classification

Consider the n × n shift matrix:

S = [ 0 1 0 … 0 0 0 1 … 0 ⋮ ⋮ ⋮ ⋱ ⋮ 0 0 0 … 1 0 0 0 … 0 ] .

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:

S ( x 1 , x 2 , … , x n ) = ( x 2 , … , x n , 0 ) .

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

[ S 1 0 … 0 0 S 2 … 0 ⋮ ⋮ ⋱ ⋮ 0 0 … S r ]

where each of the blocks S₁, S₂, ..., S_r is a shift matrix (possibly of different sizes). This form is a special case of the Jordan canonical form for matrices.

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

[ 0 1 0 0 ] .

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

{ 0 } ⊂ ker ⁡ L ⊂ ker ⁡ L 2 ⊂ … ⊂ ker ⁡ L q − 1 ⊂ ker ⁡ L q = R n

and a signature

0 = n 0 < n 1 < n 2 < … < n q − 1 < n q = n , n i = dim ⁡ ker ⁡ L i .

The signature characterizes L up to an invertible linear transformation. Furthermore, it satisfies the inequalities

n j + 1 − n j ≤ n j − n j − 1 , for all  j = 1 , … , q − 1.

Conversely, any sequence of natural numbers satisfying these inequalities is the signature of a nilpotent transformation.

Additional properties

If N is nilpotent, then I + N is invertible, where I is the n × n identity matrix. The inverse is given by

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 det ( I + t A ) = 1 for all values of t, then A is nilpotent. In fact, since p ( t ) = det ( I + t A ) − 1 is a polynomial of degree n , it suffices to have this hold for n + 1 distinct values of t .

Every singular matrix can be written as a product of nilpotent matrices.

A nilpotent matrix is a special case of a convergent matrix.

Generalizations

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

T k ( v ) = 0.

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