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Math Help - Giving an example of a nilpotent matrix

  1. #1
    MHF Contributor arbolis's Avatar
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    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  <br />
\begin{bmatrix} <br />
a & b & c\\<br />
d & e & f\\<br />
g & h & i\end{bmatrix}<br />
    and I multiplied it by itself. I got the matrix  <br />
\begin{bmatrix} <br />
a^2+bd+cg & ab+be+ch & ac+bf+ic\\<br />
ad+ed+fg & bd+e^2+fh & cd+ef+if\\<br />
ag+eh+ig & bg+eh+ih & cg+hf+i^2 \end{bmatrix}<br />


    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.
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by arbolis View Post
    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  <br />
\begin{bmatrix} <br />
a & b & c\\<br />
d & e & f\\<br />
g & h & i\end{bmatrix}<br />
    and I multiplied it by itself. I got the matrix  <br />
\begin{bmatrix} <br />
a^2+bd+cg & ab+be+ch & ac+bf+ic\\<br />
ad+ed+fg & bd+e^2+fh & cd+ef+if\\<br />
ag+eh+ig & bg+eh+ih & cg+hf+i^2 \end{bmatrix}<br />


    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}<br />
0 & a & b\\<br />
0 & 0 & c\\<br />
0 & 0 & 0<br />
\end{bmatrix}<br />
or a strictly lower triangular matrix: \begin{bmatrix}<br />
0 & 0 & 0\\<br />
d & 0 & 0\\<br />
e & f & 0<br />
\end{bmatrix}

    I'm sure there are other examples out there.

    Does this help?
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  3. #3
    MHF Contributor arbolis's Avatar
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    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.
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