Giving an example of a nilpotent matrix

• January 26th 2009, 02:01 PM
arbolis
Giving an example of a nilpotent matrix
Hi MHF,
The exercise states "Give an example of a 3x3 nilpontent matrix which is not the null matrix".
My attempt : I wrote down the matrix $
\begin{bmatrix}
a & b & c\\
d & e & f\\
g & h & i\end{bmatrix}
$

and I multiplied it by itself. I got the matrix $
\begin{bmatrix}
a^2+bd+cg & ab+be+ch & ac+bf+ic\\
ad+ed+fg & bd+e^2+fh & cd+ef+if\\
ag+eh+ig & bg+eh+ih & cg+hf+i^2 \end{bmatrix}
$

Now I must equal each entriy of the matrix to 0... that means I'll get a "nasty" system of 9 equations with 9 unknown to solve. I say "nasty" because it isn't even linear!
So I'd like to know what's the method to find such an example. Without guessing of course. I don't think my method is effective.
Thank you.
• January 26th 2009, 02:29 PM
Chris L T521
Quote:

Originally Posted by arbolis
Hi MHF,
The exercise states "Give an example of a 3x3 nilpontent matrix which is not the null matrix".
My attempt : I wrote down the matrix $
\begin{bmatrix}
a & b & c\\
d & e & f\\
g & h & i\end{bmatrix}
$

and I multiplied it by itself. I got the matrix $
\begin{bmatrix}
a^2+bd+cg & ab+be+ch & ac+bf+ic\\
ad+ed+fg & bd+e^2+fh & cd+ef+if\\
ag+eh+ig & bg+eh+ih & cg+hf+i^2 \end{bmatrix}
$

Now I must equal each entriy of the matrix to 0... that means I'll get a "nasty" system of 9 equations with 9 unknown to solve. I say "nasty" because it isn't even linear!
So I'd like to know what's the method to find such an example. Without guessing of course. I don't think my method is effective.
Thank you.

An example of a nilpotent matrix would be a strictly upper triangular matrix: $\begin{bmatrix}
0 & a & b\\
0 & 0 & c\\
0 & 0 & 0
\end{bmatrix}
$
or a strictly lower triangular matrix: $\begin{bmatrix}
0 & 0 & 0\\
d & 0 & 0\\
e & f & 0
\end{bmatrix}$

I'm sure there are other examples out there.

Does this help?
• January 26th 2009, 03:38 PM
arbolis
Of course it helps! Thank you very much.
I guess I should just memorize this result. As I've never dealt with nilpotent matrices before I had no clue.