Find the minimal polynomials and the rational forms of each of the following real matrices. \[ \begin{bmatrix} 0 & -1 & -1 \\ 1 & 0 & 0 \\ -1 & 0 & 0 \end{bmatrix}, \qquad \begin{bmatrix} c & 0 & -1 \\ 0 & c & 1 \\ -1 & 1 & c \end{bmatrix}, \qquad \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}. \]
Solution
1. \(A_1=\begin{pmatrix}0&-1&-1\\1&0&0\\-1&0&0\end{pmatrix}\) has \(\chi_{A_1}= -x(x^2+2)\), so \(m_{A_1}=x(x^2+2)\), rational form \(\operatorname{diag}(C(x^2+2),[0])\).
char poly.
2. \(A_2=\begin{pmatrix}c&0&-1\\0&c&1\\-1&1&c\end{pmatrix}\) has \(\chi=(x-c)((x-c)^2-2)\), so if \(c\neq0\) \(m=(x-c)((x-c)^2-2)\), form \(\operatorname{diag}(C((x-c)^2-2),[c])\).
char poly.
3. \(A_3=\begin{pmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{pmatrix}\) has \(\chi=x^2-2\cos\theta\,x+1\), \(m=\chi\) unless \(\sin\theta=0\), rational form \(C(\chi)\).
char poly.
4. Q.E.D.
1 and 3.