2
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Backgrounds

Theorem. For $2 \times 2$ orthogonal matrix $A$ on $\mathbb{R}$, $A$ is a Givens rotation or a Householder reflector.

Proof. Suppose we have an orthogonal matrix,

$$ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} $$

we have

$$ I = A^T A = \begin{bmatrix} a^2 +c^2 & ab + cd \\ ab + cd & b^2 + d^2 \end{bmatrix} $$

thus we have the following formulas

  1. $\Vert (a,c) \Vert_2 = 1$
  2. $\Vert (b,d) \Vert_2 = 1$
  3. $(a,c)(b,d)^T = 0$

So the solutions to the above formulas are given as

$$ \begin{aligned} (a,c) &= (\cos \theta, \sin \theta) \\[8pt] (b,d) &= \left(\cos (\theta \pm \pi/2), \sin (\theta \pm \pi/2) \right) \end{aligned} $$

So when 1) $ad - bc = 1$,

$$ \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} \cos\theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} $$

and when 2) $ad - bc = -1$,

$$ \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} \cos\theta & \sin \theta \\ \sin \theta & -\cos \theta \end{bmatrix} $$

The first case, by definition, corresponds to the Givens rotation by $\theta$.

We can also check that the second case corresponds to the the Householder reflection with respect to $v = (-\sin(\theta/2), \cos(\theta/2))$.


Question

As far as I understood, for $3 \times 3$ matrix cases, a Givens rotation is a special case of rotation, where only two axes serve as the rotational axes, and Householder reflection is also a special case of reflection, where only a one-dimensional plane serves as the axis of the reflection.

So my question is : can we generalize, that

For any $n \times n$ orthogonal matrix A on $\mathbb{R}$, $A$ is a rotation or reflection?

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  • 2
    $\begingroup$ You're missing several details in your work on the $n=2$ scenario. In general, $a,b,c,d$ are complex, and $\det A$ ranges in $\{e^{i\theta}:\theta\in[0,2\pi)\}$, not in $\{1,-1\}$. $\endgroup$ – Aweygan May 20 at 4:07
  • $\begingroup$ @Aweygan I think I need to change the whole things into $\mathbb{R}$ cases... i.e. orthogonal matrices.. $\endgroup$ – Moreblue May 20 at 4:18
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    $\begingroup$ Probably the section of Wikipedia on higher dimensions, en.wikipedia.org/wiki/Orthogonal_matrix#Higher_dimensions will give you some idea of what goes on when $n>2$, also the paragraph following that, about "primitives". $\endgroup$ – Gerry Myerson May 20 at 4:35
  • $\begingroup$ Had a look at that link, Moreblue? $\endgroup$ – Gerry Myerson May 21 at 12:46
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    $\begingroup$ I'm voting to close this question as off-topic because OP has abandoned it. $\endgroup$ – Gerry Myerson May 26 at 9:03

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