Editing Nilpotent matrix (section)

==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.