I have to find a $3$x$3$ matrix with the minimal polynomial: $x^2 -9$.
What I've tried:
If it would be a 2×2 matrix, then :
$$
C:=\begin{pmatrix}
0&9 \\ 1 & 0
\end{pmatrix}
$$
Then i tried adding columns and rows with zeros, but the minimal polynomial is now $x^3-9x$
$$
C:=\begin{pmatrix}
0&9&0 \\ 1 & 0&0 \\0&0&0
\end{pmatrix}
$$
How can i fix this?
Best Answer
Your example will not work because the eigenvalues you need for such a matrix must be $3,-3$. Your characteristic polynomial should be something like $(x-3)^2(x+3)$ or $(x+3)^2(x-3)$. Thus look for a matrix with eigenvalues as say $3,3,-3$. The simplest matrix that satisfies this will be $$A=\begin{bmatrix}3&0&0\\0&3&0\\0&0&-3\end{bmatrix}$$