Principle of distributivity

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

$N^k=0$

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 L^k = 0 for some positive integer k (and thus, L^j = 0 for all j ≥ k). Both of these concepts are special cases of a more general concept of nilpotence that applies to elements of rings.

Examples

The matrix

$M=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]$

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

$N=\left[\begin{array}{cccc}0& 2& 1& 6\\ 0& 0& 1& 2\\ 0& 0& 0& 3\\ 0& 0& 0& 0\end{array}\right]$

is nilpotent, with

$N^2=\left[\begin{array}{cccc}0& 0& 2& 7\\ 0& 0& 0& 3\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right];N^3=\left[\begin{array}{cccc}0& 0& 0& 6\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right];N^4=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right].$

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

$N=\left[\begin{array}{ccc}5& -3& 2\\ 15& -9& 6\\ 10& -6& 4\end{array}\right]$

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:

1. N is nilpotent.
2. The minimal polynomial for N is λ^k for some positive integer k ≤ n.
3. The characteristic polynomial for N is λⁿ.
4. The only (complex) eigenvalue for N is 0.
5. 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.
• The only nilpotent diagonalizable matrix is the zero matrix.

Classification

Consider the n × n shift matrix:

$S=\left[\begin{array}{ccccc}0& 1& 0& \dots & 0\\ 0& 0& 1& \dots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \dots & 1\\ 0& 0& 0& \dots & 0\end{array}\right].$

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:

$S(x_1,x_2,\dots ,x_n)=(x_2,\dots ,x_n,0).$

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

$\left[\begin{array}{cccc}{S}_1& 0& \dots & 0\\ 0& S_2& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & S_r\end{array}\right]$

where each of the blocks S₁, S₂, ..., S_r 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

$\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right].$

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

$\{0\}\subset \ker L\subset \ker L^2\subset \dots \subset \ker L^{q-1}\subset \ker L^q={\mathbb{ℝ}}^n$

and a signature

$0=n_0<n_1<n_2<\dots <n_{q-1}<n_q=n,n_i=\dim \ker L^i.$

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

$n_{j+1}-n_j\le n_j-n_{j-1},\text{for all }j=1,\dots ,q-1.$

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

$(I+N{)}^{-1}=I-N+N^2-N^3+\cdots ,$

where only finitely many terms of this sum are nonzero.

• If N is nilpotent, then

$\det (I+N)=1,$

where I denotes the n × n identity matrix. Conversely, if A is a matrix and

$\det (I+tA)=1$

for all values of t, then A is nilpotent.

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

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.

External links

• Nilpotent matrix and nilpotent transformation on PlanetMath.