Nilpotent matrix explained

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 called the index of N , sometimes the degree of N .L of a vector space such that L ^{k} = 0 for some positive integer k (and thus, L ^{j} = 0 for all j \geq k ). 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

A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} is nilpotent with index 2, since A ^{2} = 0 .

Example 2

More generally, any

n -dimensional triangular matrix with zeros along the main diagonal is nilpotent, with index \le n . For example, the matrixB = \begin{bmatrix} 0 & 2 & 1 & 6 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 0 \end{bmatrix} is nilpotent, withB ^{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}

The index of

B is therefore 4.

Example 3

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

C = \begin{bmatrix} 5 & -3 & 2 \\ 15 & -9 & 6 \\ 10 & -6 & 4 \end{bmatrix} C ^{2=\begin{bmatrix}} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} although the matrix has no zero entries.

Example 4

Additionally, any matrices of the form

\begin{bmatrix} a _{1} & a _{1} & … & a _{1} \\ a _{2} & a _{2} & … & 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}

such as

\begin{bmatrix} 5 & 5 & 5 \\ 6 & 6 & 6 \\ -11 & -11 & -11 \end{bmatrix}

or

\begin{bmatrix} 1 & 1 & 1 & 1 \\ 2 & 2 & 2 & 2 \\ 4 & 4 & 4 & 4 \\ -7 & -7 & -7 & -7 \end{bmatrix}

square to zero.

Example 5

Perhaps some of the most striking examples of nilpotent matrices are

n x n square matrices of the form:\begin{bmatrix} 2 & 2 & 2 & … & 1- n \\ n +2 & 1 & 1 & … & - n \\ 1 & n +2 & 1 & … & - n \\ 1 & 1 & n +2 & … & - n \\ \vdots & \vdots & \vdots & \ddots & \vdots \end{bmatrix}

The first few of which are:

\begin{bmatrix} 2 & -1 \\ 4 & -2 \end{bmatrix} \begin{bmatrix} 2 & 2 & -2 \\ 5 & 1 & -3 \\ 1 & 5 & -3 \end{bmatrix} \begin{bmatrix} 2 & 2 & 2 & -3 \\ 6 & 1 & 1 & -4 \\ 1 & 6 & 1 & -4 \\ 1 & 1 & 6 & -4 \end{bmatrix} \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} \ldots

These matrices are nilpotent but there are no zero entries in any powers of them less than the index.^[1]

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

n x n square matrix N with real (or complex) entries, the following are equivalent:N is nilpotent.

• The characteristic polynomial for

N is \det \left(xI - N\right) = x ^{n} .

• The minimal polynomial for

N is x ^{k} for some positive integer k \leq n .

• The only complex eigenvalue for

N 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

n x n nilpotent matrix is always less than or equal to n . For example, every 2 x 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.

See also: Jordan–Chevalley decomposition#Nilpotency criterion.

Classification

Consider the

n x n (upper) shift matrix: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}. 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,\ldots,x} _{n)} = (x _{2,\ldots,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\begin{bmatrix} S _{1} & 0 & \ldots & 0 \ 0 & S _{2} & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & S _{r} \end{bmatrix} where each of the blocks S _{1,S} _{2,\ldots,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

\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}. 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 R ^{n} naturally determines a flag of subspaces\{ 0 \} \subset \ker L \subset \ker L ^{2} \subset \ldots \subset \ker L ^{q-1} \subset \ker L ^{q} = R ^{n} and a signature0 = n _{0} < n _{1} < n _{2} < \ldots < 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 inequalitiesn _{j+1} - n _{j} \leq n _{j} - n _{j-1} , forall j = 1 ,\ldots,q -1 . Conversely, any sequence of natural numbers satisfying these inequalities is the signature of a nilpotent transformation.

Generalizations

T is locally nilpotent if for every vector v , there exists a k\in N such thatT ^{k(v)} = 0 . For operators on a finite-dimensional vector space, local nilpotence is equivalent to nilpotence.

Notes

1. Web site: Finding "nonobvious" nilpotent matrices . Mercer . Idris D. . 31 October 2005 . idmercer.com . . 5 April 2023.

External links

• Nilpotent matrix and nilpotent transformation on PlanetMath.