Math 511: Linear Algebra
Project Nilpotent Matrices
Nilpotent Matrices¶
Let $A$ be a nonzero square matrix. Is it possible that a $A^k=O$ 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 $A\neq 0$, $A^2\neq 0$, $\dots$, $A^{k-1}\neq 0$, but $A^k=0$.
1.¶
The matrix in the introduction is nilpotent. What is its index?
2.¶
Use a software program or a graphing utility or compute integer powers of the following matrices to determine which matrices below are nilpotent and find their indices. Include in your pdf screen shots 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 a $4\times 4$ matrices of index 2, 3 and 4. Show that they are nilpotent of the correct index.
5.¶
Are nilpotent matrices invertible? Prove your answer.
6.¶
Show that if $A$ is nilpotent, then $I-A$ is invertible.