Nilpotent matrix

Short description: Mathematical concept in algebra

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

[math]\displaystyle{ N^k = 0\, }[/math]

for some positive integer [math]\displaystyle{ k }[/math]. The smallest such [math]\displaystyle{ k }[/math] is called the index of [math]\displaystyle{ N }[/math], sometimes the degree of [math]\displaystyle{ N }[/math].

More generally, a nilpotent transformation is a linear transformation [math]\displaystyle{ L }[/math] of a vector space such that [math]\displaystyle{ L^k = 0 }[/math] for some positive integer [math]\displaystyle{ k }[/math] (and thus, [math]\displaystyle{ L^j = 0 }[/math] for all [math]\displaystyle{ j \geq k }[/math]). 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

[math]\displaystyle{ A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} }[/math]

is nilpotent with index 2, since [math]\displaystyle{ A^2 = 0 }[/math].

Example 2

More generally, any [math]\displaystyle{ n }[/math]-dimensional triangular matrix with zeros along the main diagonal is nilpotent, with index [math]\displaystyle{ \le n }[/math]. For example, the matrix

[math]\displaystyle{ B=\begin{bmatrix} 0 & 2 & 1 & 6\\ 0 & 0 & 1 & 2\\ 0 & 0 & 0 & 3\\ 0 & 0 & 0 & 0 \end{bmatrix} }[/math]

is nilpotent, with

[math]\displaystyle{ B^2=\begin{bmatrix} 0 & 0 & 2 & 7\\ 0 & 0 & 0 & 3\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} ;\ B^3=\begin{bmatrix} 0 & 0 & 0 & 6\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} ;\ B^4=\begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix}. }[/math]

The index of [math]\displaystyle{ B }[/math] is therefore 4.

Example 3

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

[math]\displaystyle{ C=\begin{bmatrix} 5 & -3 & 2 \\ 15 & -9 & 6 \\ 10 & -6 & 4 \end{bmatrix} \qquad C^2=\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} }[/math]

although the matrix has no zero entries.

Example 4

Additionally, any matrices of the form

[math]\displaystyle{ \begin{bmatrix} a_1 & a_1 & \cdots & a_1 \\ a_2 & a_2 & \cdots & a_2 \\ \vdots & \vdots & \ddots & \vdots \\ -a_1-a_2-\ldots-a_{n-1} & -a_1-a_2-\ldots-a_{n-1} & \ldots & -a_1-a_2-\ldots-a_{n-1} \end{bmatrix} }[/math]

such as

[math]\displaystyle{ \begin{bmatrix} 5 & 5 & 5 \\ 6 & 6 & 6 \\ -11 & -11 & -11 \end{bmatrix} }[/math]

or

[math]\displaystyle{ \begin{bmatrix} 1 & 1 & 1 & 1 \\ 2 & 2 & 2 & 2 \\ 4 & 4 & 4 & 4 \\ -7 & -7 & -7 & -7 \end{bmatrix} }[/math]

square to zero.

Example 5

Perhaps some of the most striking examples of nilpotent matrices are [math]\displaystyle{ n\times n }[/math] square matrices of the form:

[math]\displaystyle{ \begin{bmatrix} 2 & 2 & 2 & \cdots & 1-n \\ n+2 & 1 & 1 & \cdots & -n \\ 1 & n+2 & 1 & \cdots & -n \\ 1 & 1 & n+2 & \cdots & -n \\ \vdots & \vdots & \vdots & \ddots & \vdots \end{bmatrix} }[/math]

The first few of which are:

[math]\displaystyle{ \begin{bmatrix} 2 & -1 \\ 4 & -2 \end{bmatrix} \qquad \begin{bmatrix} 2 & 2 & -2 \\ 5 & 1 & -3 \\ 1 & 5 & -3 \end{bmatrix} \qquad \begin{bmatrix} 2 & 2 & 2 & -3 \\ 6 & 1 & 1 & -4 \\ 1 & 6 & 1 & -4 \\ 1 & 1 & 6 & -4 \end{bmatrix} \qquad \begin{bmatrix} 2 & 2 & 2 & 2 & -4 \\ 7 & 1 & 1 & 1 & -5 \\ 1 & 7 & 1 & 1 & -5 \\ 1 & 1 & 7 & 1 & -5 \\ 1 & 1 & 1 & 7 & -5 \end{bmatrix} \qquad \ldots }[/math]

These matrices are nilpotent but there are no zero entries in any powers of them less than the index.

Example 6

Consider the linear space of polynomials of a bounded degree. The derivative operator is a linear map. We know that applying the derivative to a polynomial decreases its degree by one, so when applying it iteratively, we will eventually obtain zero. Therefore, on such a space, the derivative is representable by a nilpotent matrix.

Characterization

For an [math]\displaystyle{ n \times n }[/math] square matrix [math]\displaystyle{ N }[/math] with real (or complex) entries, the following are equivalent:

• [math]\displaystyle{ N }[/math] is nilpotent.
• The characteristic polynomial for [math]\displaystyle{ N }[/math] is [math]\displaystyle{ \det \left(xI - N\right) = x^n }[/math].
• The minimal polynomial for [math]\displaystyle{ N }[/math] is [math]\displaystyle{ x^k }[/math] for some positive integer [math]\displaystyle{ k \leq n }[/math].
• The only complex eigenvalue for [math]\displaystyle{ N }[/math] is 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 index of an [math]\displaystyle{ n \times n }[/math] nilpotent matrix is always less than or equal to [math]\displaystyle{ n }[/math]. For example, every [math]\displaystyle{ 2 \times 2 }[/math] 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.

See also: Jordan–Chevalley decomposition.

Classification

Consider the [math]\displaystyle{ n \times n }[/math] (upper) shift matrix:

[math]\displaystyle{ S = \begin{bmatrix} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \ldots & 1 \\ 0 & 0 & 0 & \ldots & 0 \end{bmatrix}. }[/math]

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:

[math]\displaystyle{ S(x_1,x_2,\ldots,x_n) = (x_2,\ldots,x_n,0). }[/math]

This matrix is nilpotent with degree [math]\displaystyle{ n }[/math], and is the canonical nilpotent matrix.

Specifically, if [math]\displaystyle{ N }[/math] is any nilpotent matrix, then [math]\displaystyle{ N }[/math] is similar to a block diagonal matrix of the form

[math]\displaystyle{ \begin{bmatrix} S_1 & 0 & \ldots & 0 \\ 0 & S_2 & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & S_r \end{bmatrix} }[/math]

where each of the blocks [math]\displaystyle{ S_1,S_2,\ldots,S_r }[/math] 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

[math]\displaystyle{ \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}. }[/math]

That is, if [math]\displaystyle{ N }[/math] 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 [math]\displaystyle{ L }[/math] on [math]\displaystyle{ \mathbb{R}^n }[/math] naturally determines a flag of subspaces

[math]\displaystyle{ \{0\} \subset \ker L \subset \ker L^2 \subset \ldots \subset \ker L^{q-1} \subset \ker L^q = \mathbb{R}^n }[/math]

and a signature

[math]\displaystyle{ 0 = n_0 \lt n_1 \lt n_2 \lt \ldots \lt n_{q-1} \lt n_q = n,\qquad n_i = \dim \ker L^i. }[/math]

The signature characterizes [math]\displaystyle{ L }[/math] up to an invertible linear transformation. Furthermore, it satisfies the inequalities

[math]\displaystyle{ n_{j+1} - n_j \leq n_j - n_{j-1}, \qquad \mbox{for all } j = 1,\ldots,q-1. }[/math]

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

Additional properties

• If [math]\displaystyle{ N }[/math] is nilpotent of index [math]\displaystyle{ k }[/math] , then [math]\displaystyle{ I+N }[/math] and [math]\displaystyle{ I-N }[/math] are invertible, where [math]\displaystyle{ I }[/math] is the [math]\displaystyle{ n \times n }[/math] identity matrix. The inverses are given by

  [math]\displaystyle{ \begin{align} (I + N)^{-1} &= \displaystyle\sum^k_{m=0}\left(-N\right)^m = I - N + N^2 - N^3 + N^4 - N^5 + N^6 - N^7 + \cdots +(-N)^k \\ (I - N)^{-1} &= \displaystyle\sum^k_{m=0}N^m = I + N + N^2 + N^3 + N^4 + N^5 + N^6 + N^7 + \cdots + N^k \\ \end{align} }[/math]

• If [math]\displaystyle{ N }[/math] is nilpotent, then
  [math]\displaystyle{ \det (I + N) = 1. }[/math]

  Conversely, if [math]\displaystyle{ A }[/math] is a matrix and

  [math]\displaystyle{ \det (I + tA) = 1\!\, }[/math]

  for all values of [math]\displaystyle{ t }[/math], then [math]\displaystyle{ A }[/math] is nilpotent. In fact, since [math]\displaystyle{ p(t) = \det (I + tA) - 1 }[/math] is a polynomial of degree [math]\displaystyle{ n }[/math], it suffices to have this hold for [math]\displaystyle{ n+1 }[/math] distinct values of [math]\displaystyle{ t }[/math].
• 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 [math]\displaystyle{ T }[/math] is locally nilpotent if for every vector [math]\displaystyle{ v }[/math], there exists a [math]\displaystyle{ k\in\mathbb{N} }[/math] such that

[math]\displaystyle{ T^k(v) = 0.\!\, }[/math]

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

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, https://archive.org/details/firstcourseinlin0000beau 
• 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: John Wiley & Sons 

External links

• Nilpotent matrix and nilpotent transformation on PlanetMath.