# Giving an example of a nilpotent matrix

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

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

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: $\displaystyle \begin{bmatrix} 0 & a & b\\ 0 & 0 & c\\ 0 & 0 & 0 \end{bmatrix}$ or a strictly lower triangular matrix: $\displaystyle \begin{bmatrix} 0 & 0 & 0\\ d & 0 & 0\\ e & f & 0 \end{bmatrix}$

I'm sure there are other examples out there.

Does this help?
• Jan 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.