I want to find the minimal polynomial (the monic polynomial of least positive degree that annihilates the matrix) of the following matrix:
$$\begin{pmatrix}
0 & 1 & 1 & 0\\
-1 & 0 & 0 & 1\\
-1 & 0 & 0 & 1\\
0 & -1 & -1 & 0
\end{pmatrix}
$$
I know how to use the characteristic polynomial to find the minimal polynomial, but is there a way to get around that? It's not quite symmetric, so I can't use the trick that applies to symmetric matrices.
Thank you.
Best Answer
The matrix is skew-symmetric and one can write it as
$$M=\begin{bmatrix}r_1\\r_2\\r_2\\-r_1\end{bmatrix}$$
where the rows $r_1$ and $r_2$ are immediate from the definition. Since it is skew-symmetric, then we also have that
$$M=\begin{bmatrix}-r_1^T & -r_2^T&-r_2^T&r_1^T\end{bmatrix}.$$
We have that $r_1^Tr_2=0$ and $r_1^Tr_1=r_2^Tr_2=2$ together with $r_1M=2r_2$ and $r_2M=-2r_1$.
Therefore,
$$M^2=2\begin{bmatrix}r_2\\-r_1\\-r_1\\-r_2\end{bmatrix}\ \mathrm{and}\ M^3=-4\begin{bmatrix}r_1\\r_2\\r_2\\r_1\end{bmatrix}=-4M.$$
Therefore, the minimal polynomial is $p(x)=x^3+4x=x(x^2+4)$.