# Orthogonal matrix, translations, and fixed points

Is it true that $Ax+b$ where $A^\dagger A=I=AA^\dagger$ and $A$ is an $n\times n$ real matrix, $x,b\in \mathbb R^n$ must have either a fixed point or a fixed $n-1$ hyperplane? If not, is it true for a smaller $n$? Thanks.

[Edit:] Perhaps if we only consider the small $n$ cases such as $n=2,3$, the question might be more friendly and hopefully I will get a response? Thanks.

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Rotate and translate along the axis of rotation. – Louis Feb 3 '12 at 23:46
@Louis: Is there an algebraic proof? And why is the translation necessarily along the axis of rotation? $A$ could be a reflection too, no? – Charlie Feb 4 '12 at 0:06

If $A$ is an $n$ by $n$ orthogonal matrix and $b\in {\Bbb R}^n$, then the map $\phi: x \mapsto Ax+b$ need not have any fixed points in ${\Bbb R}^n$. For example, $A$ could be the identity and $b\ne 0$.

In general, by a change of coordinate system, $A$ can be made block-diagonal, where each block is either the size-1 block $1$, the size-1 block $-1$, or a size-2 block which is a rotation matrix $R_\theta=\left(\cos \theta\ \ \sin \theta\atop-\sin \theta\ \ \cos \theta\right)$, $\theta\in(0,\pi)$. Suppose that there are $k$ blocks $1$, $\ell$ blocks $-1$, and $m$ blocks $R_{\theta_1}$, $\ldots$, $R_{\theta_m}$, where $k+\ell+2m=n$. Breaking up ${\Bbb R}^n$ according to these block types, we can write ${\Bbb R}^n=U\oplus V\oplus W$, where $\dim U=k$, $\dim V=\ell$, $\dim W=2m$. We also have projections $\pi_U$, $\pi_V$, and $\pi_W$ from ${\Bbb R}^n$ to $U$, $V$, and $W$. We can then take the equation $$x=\phi(x)=Ax+b \qquad (*)$$ and project it, getting $$\pi_U(x)=\pi_U(x)+\pi_U(b), \qquad (1)$$ $$\pi_V(x)=-\pi_V(x)+\pi_V(b), \qquad (2)$$ $$\pi_W(x)={\rm diag}(R_{\theta_1},\ldots,R_{\theta_m}) \pi_W(x)+\pi_W(b). \qquad (3)$$ (1) will never be satisfied if $\pi_U(b)\ne 0$ but is always satisfied if $\pi_U(b)=0$. (2) and (3) always have unique solutions for $\pi_V(x)$ and $\pi_W(x)$. So, (*) will have no solutions if $\pi_U(b)\ne 0$, but will have an affine space of solutions of dimension $k$ if $\pi_U(b)=0$.

In dimensions 2 and 3, $\phi$ has a special name depending on $k$, $l$, $m$, and $\pi_U(b)$:

Dimension 2:

• $k=2$, $\ell=0$, $m=0$, $b=0$: identity (the whole plane fixed)

• $k=2$, $\ell=0$, $m=0$, $b\ne 0$: translation (no fixed points)

• $k=1$, $\ell=1$, $m=0$, $\pi_U(b)=0$: reflection (line of fixed points)

• $k=1$, $\ell=1$, $m=0$, $\pi_U(b)\ne 0$: glide reflection (no fixed points)

• $k=0$, $\ell=2$, $m=0$: inversion in a point (rotation by 180°; one fixed point)

• $k=0$, $\ell=0$, $m=1$: rotation (one fixed point)

Dimension 3:

• $k=3$, $\ell=0$, $m=0$, $b=0$: identity (all of space fixed)

• $k=3$, $\ell=0$, $m=0$, $b\ne 0$: translation (no fixed points)

• $k=2$, $\ell=1$, $m=0$, $\pi_U(b)=0$: reflection (a plane of fixed points)

• $k=2$, $\ell=1$, $m=0$, $\pi_U(b)\ne 0$: glide reflection (no fixed points)

• $k=1$, $\ell=2$, $m=0$, $\pi_U(b)=0$: 180° rotation (reflection in a line; an axis of fixed points)

• $k=1$, $\ell=0$, $m=1$, $\pi_U(b)=0$: rotation (an axis of fixed points)

• $k=1$, $\ell=2$, $m=0$, $\pi_U(b)\ne 0$: 180° rotation plus translation along rotation axis (screw motion; no fixed points)

• $k=1$, $\ell=0$, $m=1$, $\pi_U(b)\ne 0$: rotation plus translation along rotation axis (screw motion; no fixed points)

• $k=0$, $\ell=3$, $m=0$: inversion in a point (180° rotation in an axis plus reflection in a plane normal to this axis; one fixed point)

• $k=0$, $\ell=1$, $m=1$: rotation in an axis plus reflection in a plane normal to this axis (rotatory reflection or rotoreflection; one fixed point)

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