Editing Nilpotent matrix

{{Short description|Mathematical concept in algebra}}
In [[linear algebra]], a '''nilpotent matrix''' is a [[square matrix]] ''N'' such that
:<math>N^k = 0\,</math>
for some positive [[integer]] <math>k</math>.  The smallest such <math>k</math> is called the '''index''' of <math>N</math>,<ref>{{harvtxt|Herstein|1975|p=294}}</ref> sometimes the '''degree''' of <math>N</math>.

More generally, a '''nilpotent transformation''' is a [[linear transformation]] <math>L</math> of a [[vector space]] such that <math>L^k = 0</math> for some positive integer <math>k</math> (and thus, <math>L^j = 0</math> for all <math>j \geq k</math>).<ref>{{harvtxt|Beauregard|Fraleigh|1973|p=312}}</ref><ref>{{harvtxt|Herstein|1975|p=268}}</ref><ref>{{harvtxt|Nering|1970|p=274}}</ref> Both of these concepts are special cases of a more general concept of [[nilpotent|nilpotence]] that applies to elements of [[ring (algebra)|rings]].

==Examples==
===Example 1===
The matrix
:<math> 
A = \begin{bmatrix} 
0 & 1 \\
0 & 0 
\end{bmatrix}
</math>
is nilpotent with index 2, since <math>A^2 = 0</math>.

===Example 2===
More generally, any <math>n</math>-dimensional [[triangular matrix]] with zeros along the [[main diagonal]] is nilpotent, with index <math>\le n</math> {{Citation needed|date=November 2022}}. For example, the matrix
:<math> 
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>
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>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> 
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>
\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>
\begin{bmatrix}
5 & 5 & 5 \\
6 & 6 & 6 \\
-11 & -11 & -11
\end{bmatrix}
</math>

or 

:<math>\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>n\times n</math> square matrices of the form:

:<math>\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>\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.<ref name="Mercer2005">{{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
}}</ref>

===Example 6===
Consider the linear space of [[polynomial]]s 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==
{{Unreferenced section|date=May 2018}}
For an <math>n \times n</math> square matrix <math>N</math> with [[real number|real]] (or [[complex number|complex]]) entries, the following are equivalent:
* <math>N</math> is nilpotent.
* The [[characteristic polynomial]] for <math>N</math> is <math>\det \left(xI - N\right) = x^n</math>.
* The [[minimal polynomial (linear algebra)|minimal polynomial]] for <math>N</math> is <math>x^k</math> for some positive integer <math>k \leq n</math>.
* The only complex eigenvalue for <math>N</math> is 0.
The last theorem holds true for matrices over any [[field (mathematics)|field]] of characteristic 0 or sufficiently large characteristic. (cf. [[Newton's identities]])

This theorem has several consequences, including:
* The index of an <math>n \times n</math> nilpotent matrix is always less than or equal to <math>n</math>.  For example, every <math>2 \times 2</math> nilpotent matrix squares to zero.
* The [[determinant]] and [[trace (linear algebra)|trace]] of a nilpotent matrix are always zero. Consequently, a nilpotent matrix cannot be [[invertible matrix|invertible]].
* The only nilpotent [[diagonalizable matrix]] is the zero matrix.

See also: [[Jordan–Chevalley decomposition#Nilpotency criterion]].

==Classification==
Consider the <math>n \times n</math> (upper) [[shift matrix]]:
:<math>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>S(x_1,x_2,\ldots,x_n) = (x_2,\ldots,x_n,0).</math><ref>{{harvtxt|Beauregard|Fraleigh|1973|p=312}}</ref>
This matrix is nilpotent with degree <math>n</math>, and is the [[Canonical form|canonical]] nilpotent matrix.

Specifically, if <math>N</math> is any nilpotent matrix, then <math>N</math> is [[matrix similarity|similar]] to a [[block diagonal matrix]] of the form
:<math> \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>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.<ref>{{harvtxt|Beauregard|Fraleigh|1973|pp=312,313}}</ref>

For example, any nonzero 2&nbsp;&times;&nbsp;2 nilpotent matrix is similar to the matrix
:<math> \begin{bmatrix} 
   0 & 1 \\
   0 & 0
\end{bmatrix}. </math>
That is, if <math>N</math> is any nonzero 2&nbsp;&times;&nbsp;2 nilpotent matrix, then there exists a basis '''b'''<sub>1</sub>,&nbsp;'''b'''<sub>2</sub> such that ''N'''''b'''<sub>1</sub>&nbsp;=&nbsp;0 and ''N'''''b'''<sub>2</sub>&nbsp;=&nbsp;'''b'''<sub>1</sub>.

This [[classification theorem]] holds for matrices over any [[field (mathematics)|field]].  (It is not necessary for the field to be algebraically closed.)

==Flag of subspaces==

A nilpotent transformation <math>L</math> on <math>\mathbb{R}^n</math> naturally determines a [[flag (linear algebra)|flag]] of subspaces
:<math> \{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> 0 = n_0 < n_1 < n_2 < \ldots < n_{q-1} < n_q = n,\qquad n_i = \dim \ker L^i. </math>

The signature characterizes <math>L</math> [[up to]] an invertible [[linear transformation]]. Furthermore, it satisfies the inequalities
:<math> 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==
{{unordered list
| If <math>N</math> is nilpotent of index <math>k</math> , then <math>I+N</math> and <math>I-N</math> are [[invertible matrix|invertible]], where <math>I</math> is the <math>n \times n</math> [[identity matrix]]. The inverses are given by
: <math>\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>N</math> is nilpotent, then
: <math>\det (I + N) = 1.</math>

Conversely, if <math>A</math> is a matrix and
: <math>\det (I + tA) = 1\!\,</math>
for all values of <math>t</math>, then <math>A</math> is nilpotent. In fact, since <math>p(t) = \det (I + tA) - 1</math> is a polynomial of degree <math>n</math>, it suffices to have this hold for <math>n+1</math> distinct values of <math>t</math>.
| Every [[singular matrix]] can be written as a product of nilpotent matrices.<ref>R. Sullivan, Products of nilpotent matrices, ''Linear and Multilinear Algebra'', Vol. 56, No. 3</ref>
| A nilpotent matrix is a special case of a [[convergent matrix]].
}}

==Generalizations==
A [[linear operator]] <math>T</math> is '''locally nilpotent''' if for every vector <math>v</math>, there exists a <math>k\in\mathbb{N}</math> such that
:<math>T^k(v) = 0.\!\,</math>
For operators on a finite-dimensional vector space, local nilpotence is equivalent to nilpotence.

==Notes==
<references />

==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 = [[John Wiley & Sons|Wiley]] | location = New York | lccn = 76091646 }}

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

{{Matrix classes}}

[[Category:Matrices (mathematics)]]