Chapter 3

Show that each of the following matrices is orthogonal and find the rotation and/or reflection it produces as an operator acting on vectors. If a rotation, find the axis and angle; if a reflection, find the reflecting plane and the rotation, if any, about the normal to that plane. $$\frac{1}{9}\left(\begin{array}{rrr}-1& 8& 4\\ -4& -4& 7\\ -8& 1& -4\end{array}\right)$$

Show that if $D$ is a diagonal matrix, then $D^n$ is the diagonal matrix with elements equal to the $n^{\text{th }}$ power of the elements of $\mathrm{D}$. Also show that if $\mathrm{D}={\mathrm{C}}^{-1}\mathrm{MC}$, then ${\mathrm{D}}^n={\mathrm{C}}^{-1}{\mathrm{M}}^n\mathrm{C},$ so ${\mathrm{M}}^n={\mathrm{CD}}^n{\mathrm{C}}^{-1}.$ Hint: For $n=2,{\left({\mathrm{C}}^{-1}\mathrm{MC}\right)}^2={\mathrm{C}}^{-1}{\mathrm{MCC}}^{-1}\mathrm{MC}$ what is ${\mathrm{CC}}^{-1}?$

Show that if matrices $F$ and $G$ can be diagonalized by the same $C$ matrix, then they commute. Hint: Do diagonal matrices commute?