Show isometry and find geometric meaning

[schniefen]

TL;DR

With respect to an orthonormal basis in 3-space, show that the matrix of $F$ given below is an isometry and find its geometric meaning.

The matrix $A$ in question is

$\dfrac{1}{3} \left(\begin{array}{rrr} -2 & 1 & -2 \\ -2 & -2 & 1 \\ 1 & -2 & -2 \end{array}\right)$
​

One can easily verify that $AA^t=I$, hence an isometry. To find its geometric meaning, one can proceed to find $U=\text{ker} \ (F-I)=\text{ker} \ (A-I)$, i.e. $(A-I)\textbf{x}=\textbf{0}$. This yields $\textbf{x}=\textbf{0}$. Put $G=-F$ and find $U=\text{ker} \ (G-I)=\text{ker} \ (A+I)$ instead, which yields $\textbf{x}=t(1,1,1)$. Thus $G$ is a rotation about the line $t(1,1,1)$. To determine the orientation of the rotation, one can define $\textbf{w}=(1,1,1)$, $\textbf{u}=(1,-1,0)$ (any vector perpendicular to $\textbf{w}$) and $\textbf{v}=A\textbf{u}=(-1,0,1)$. Evaluating the determinant with the columns given by $\textbf{u},\textbf{v},\textbf{w}$ shows that they are negatively oriented and thus the rotation is clockwise seen from the point $(1,1,1)$ down on the origin. The angle of rotation is deduced from the dot product of $\textbf{u}$ and $\textbf{v}$, which is $\frac{2\pi}{3}$.

This is the geometric meaning of $G$. What is that of $F$ and its matrix? The answer given is a rotation $\frac{\pi}{3}$ anti-clockwise from the point $(1,1,1)$ followed by reflection in the origin. Since $G=-F$, isn't $F$ the rotation of $G$ followed by reflection in the origin? That is, a clockwise rotation $\frac{2\pi}{3}$ from the point $(1,1,1)$ looking down on the origin followed by reflection. The answer seems to reverse this.

 

[member 587159]

If $G = -F$, let $v \in \mathbb{R}^3$. Geometrically, $Fv = -Gv$ and thus $v$ transforms first under $G$ and the result $Gv$ gets an opposite sign, meaning that we reflect the vector $Gv$ around the origin.

But the order shouldn't matter actually: since the maps involved are linear, we also have $Fv = G(-v)$ and then $F$ first reflects the vector $v$ around the origin and then performs a rotation.

 

[schniefen]

Okay, so the answers are equivalent?

 

[schniefen]

Could it be that the answer tries to describe the transformation $G$ in terms of $F$? Then it would make sense. However, that wouldn’t be the geometric meaning of $F$ and its matrix.