I need some help with this problem:
We know that $T(x_1,x_2,x_3)=(x_2,x_3,x_1)$. We suspect that it is a rotation matrix, to be sure, we need to determine the rotation axis and the rotation angle.
I first tried to see what the linear transformation did to the basis vectors.
$$T(1,0,0)=(0,0,1)$$ $$T(0,1,0)=(1,0,0)$$ $$T(0,0,1)=(0,1,0)$$ With this informatio, I wrote the matrix that represents the linear transformation: $$A=\left[\begin{matrix}
0&1&0\\0&0&1\\1&0&0
\end{matrix}\right]$$
After that, I don't know how to determine the angle and the axis, can you help me?
Best Answer
A vector on the axis of the rotation matrix $A$ will satisfy $Av=v$, that is $(A-I)v=0$. In this example, $$A-I=\pmatrix{-1&1&0\\0&-1&1\\1&0&-1}.$$ and $(A-I)\pmatrix{x\\y\\z}=\pmatrix{0\\0\\0}$ iff $x=y=z$, so the axis is the line joining the origin to $(1,1,1)$.
In this example, $A^3=I$, so the angle of rotation is a third of a turn: $\frac{2\pi}3$.
More generally if $\theta$ is the rotation angle of a rotation matrix $A$, then the trace of $A$ is $1+2\cos\theta$, so the angle can easily be found from this.