Nilpotent Matrices¶
Let $A$ be a nonzero square matrix. Is it possible that $A^k=O$ for some positive integer power $k$? For example, find $A^3$ for
$$ \begin{align*} A &= \begin{bmatrix} 0 & 1 & 2 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix} \\ \\ A^2 &= \begin{bmatrix} 0 & 1 & 2 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}\begin{bmatrix} 0 & 1 & 2 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix} \\ &= \begin{bmatrix} {\color{red}0} & {\color{blue}1} & {\color{green}2} \\ {\color{red}0} & {\color{blue}0} & {\color{green}1} \\ {\color{red}0} & {\color{blue}0} & {\color{green}0} \end{bmatrix}\begin{bmatrix} {\color{red}0} & {\color{red}1} & {\color{red}2} \\ {\color{blue}0} & {\color{blue}0} & {\color{blue}1} \\ {\color{green}0} &{\color{green}0} & {\color{green}0} \end{bmatrix} = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \\ \\ A^3 &= AA^2 = \begin{bmatrix} 0 & 1 & 2 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}\begin{bmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \end{align*} $$
Definition¶
A matrix is called nilpotent of index k when for positive integers $1\le m\lt k$, $A^m\neq O$, but $A^k=O$.
1.¶
The matrix in the introduction is nilpotent. What is its index?
2.¶
Use a software program or a graphing utility to compute integer powers of the following matrices to determine which matrices below are nilpotent and find their indices. Include in your PDF screenshots of your work if you use a software program or utility.
(a) $\begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix}$
(b) $\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{bmatrix}$
(c) $\begin{bmatrix} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{bmatrix}$
(d) $\begin{bmatrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 1 & 1 & 0 \end{bmatrix}$
3.¶
Find a nilpotent $2\times 2$ matrix of index 2. Show that it is nilpotent of the correct index.
4.¶
Find $4\times 4$ matrices of index 2, 3, and 4. Show that they are nilpotent of the correct index.
5.¶
Find a nilpotent matrix $A$ of index 5. Show that $A^4\neq O$, but $A^5=O$.
6.¶
Prove that if $n\times n$ matrix $A$ is nilpotent with index $k$, then for all positive integers $m > k$, $A^m=O$.
7.¶
When $A$ is nilpotent, what can you say about $A^\mathsf{T}$? Prove your answer.
8.¶
Show that if $A$ is nilpotent, then $I-A$ is invertible.
9.¶
Prove that if $A$ is non-singular, then it cannot be nilpotent.
10.¶
Prove that if $A$ is nilpotent, then it cannot be invertible.