nilpotent matrix

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 $N$ ,{{harvtxt|Herstein|1975|p=294}} sometimes 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 $j \geq k$ ).{{harvtxt|Beauregard|Fraleigh|1973|p=312}}{{harvtxt|Herstein|1975|p=268}}{{harvtxt|Nering|1970|p=274}} 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 $\le n$ {{Citation needed|date=November 2022}}. For example, the matrix

: $B=\begin{bmatrix}
0 & 2 & 1 & 6\\
0 & 0 & 1 & 2\\
0 & 0 & 0 & 3\\
0 & 0 & 0 & 0
\end{bmatrix}$

is nilpotent, with

: $B^2=\begin{bmatrix}
0 & 0 & 2 & 7\\
0 & 0 & 0 & 3\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{bmatrix}
;\
B^3=\begin{bmatrix}
0 & 0 & 0 & 6\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{bmatrix}
;\
B^4=\begin{bmatrix}
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{bmatrix}$

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,

: $C=\begin{bmatrix}
5 & -3 & 2 \\
15 & -9 & 6 \\
10 & -6 & 4
\end{bmatrix}
\qquad
C^2=\begin{bmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{bmatrix}$

although the matrix has no zero entries.

=Example 4=

Additionally, any matrices of the form

: $\begin{bmatrix}
a_1 & a_1 & \cdots & a_1 \\
a_2 & a_2 & \cdots & a_2 \\
\vdots & \vdots & \ddots & \vdots \\
-a_1-a_2-\ldots-a_{n-1} & -a_1-a_2-\ldots-a_{n-1} & \ldots & -a_1-a_2-\ldots-a_{n-1}
\end{bmatrix}$

such as

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

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

square to zero.

=Example 5=

Perhaps some of the most striking examples of nilpotent matrices are $n\times n$ square matrices of the form:

: $\begin{bmatrix}
2 & 2 & 2 & \cdots & 1-n \\
n+2 & 1 & 1 & \cdots & -n \\
1 & n+2 & 1 & \cdots & -n \\
1 & 1 & n+2 & \cdots & -n \\
\vdots & \vdots & \vdots & \ddots & \vdots
\end{bmatrix}$

The first few of which are:

: $\begin{bmatrix}
2 & -1 \\
4 & -2
\end{bmatrix}
\qquad
\begin{bmatrix}
2 & 2 & -2 \\
5 & 1 & -3 \\
1 & 5 & -3
\end{bmatrix}
\qquad
\begin{bmatrix}
2 & 2 & 2 & -3 \\
6 & 1 & 1 & -4 \\
1 & 6 & 1 & -4 \\
1 & 1 & 6 & -4
\end{bmatrix}
\qquad
\begin{bmatrix}
2 & 2 & 2 & 2 & -4 \\
7 & 1 & 1 & 1 & -5 \\
1 & 7 & 1 & 1 & -5 \\
1 & 1 & 7 & 1 & -5 \\
1 & 1 & 1 & 7 & -5
\end{bmatrix}
\qquad
\ldots$

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

|url=http://www.idmercer.com/nilpotent.pdf

|title=Finding "nonobvious" nilpotent matrices

|last1=Mercer

|first1=Idris D.

|date=31 October 2005

|website=idmercer.com

|publisher=self-published; personal credentials: PhD Mathematics, Simon Fraser University

|access-date=5 April 2023

}}

=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 $n \times n$ 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 $n \times n$ nilpotent matrix is always less than or equal to $n$ . For example, every $2 \times 2$ 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 $n \times n$ (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:

: $S(x_1,x_2,\ldots,x_n) = (x_2,\ldots,x_n,0).$ {{harvtxt|Beauregard|Fraleigh|1973|p=312}}

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.{{harvtxt|Beauregard|Fraleigh|1973|pp=312,313}}

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 $\mathbb{R}^n$ naturally determines a flag of subspaces

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

and a signature

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

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

: $n_{j+1} - n_j \leq n_j - n_{j-1}, \qquad \mbox{for all } j = 1,\ldots,q-1.$

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

Additional properties

{{unordered list

| If $N$ is nilpotent of index $k$ , then $I+N$ and $I-N$ are invertible, where $I$ is the $n \times n$ identity matrix. The inverses are given by

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

| If $N$ is nilpotent, then

: $\det (I + N) = 1.$

Conversely, if $A$ is a matrix and

: $\det (I + tA) = 1\!\,$

for all values of $t$ , then $A$ is nilpotent. In fact, since $p(t) = \det (I + tA) - 1$ 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.R. Sullivan, Products of nilpotent matrices, Linear and Multilinear Algebra, Vol. 56, No. 3

| 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 $k\in\mathbb{N}$ such that

: $T^k(v) = 0.\!\,$

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

Notes

References

{{citation | first1 = Raymond A. | last1 = Beauregard | first2 = John B. | last2 = Fraleigh | year = 1973 | isbn = 0-395-14017-X | title = A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields | publisher = Houghton Mifflin Co. | location = Boston | url-access = registration | url = https://archive.org/details/firstcourseinlin0000beau }}
{{citation | first = I. N. | last = Herstein | author-link= Israel Nathan Herstein | year = 1975 | title = Topics In Algebra | edition= 2nd | publisher = John Wiley & Sons }}
{{ citation | first1 = Evar D. | last1 = Nering | year = 1970 | title = Linear Algebra and Matrix Theory | edition = 2nd | publisher = Wiley | location = New York | lccn = 76091646 }}

External links

[http://planetmath.org/nilpotentmatrix Nilpotent matrix] and [http://planetmath.org/nilpotenttransformation nilpotent transformation] on PlanetMath.

Category:Matrices (mathematics)