Prove all Elements of O(2,R) have form of Rotation Matrix

[MxwllsPersuasns]

Homework Statement

Show that every matrix A ∈ O(2, R) is of the form R(α) = cos α − sin α sin α cos α (this is the 2d rotation matrix -- I can't make it in matrix format) or JR(α). Interpret the maps x → R(α)x and x → JR(α)x for x ∈ R 2

Homework Equations

The Attempt at a Solution

So I know that O(2,R) is the group of isometries (distance-preserving transformations) of the real plane. I know that the orthogonal group can be represented in this way (with the binary operation being composition of transformations) or it can be represented in the following way:

It is the group of 2-D Real Orthogonal Matrices with matrix multiplication being the Binary operation. With an orthogonal matrix being a matrix whose inverse equals its transpose.

I wasn't able to find any lists of the elements A of O(2,R) or O(n) on any website so by that I imagine we must use the general properties to show that any element that satisfies those particular properties must be of the form R(α). I'm unsure how to go about doing this exactly though.. I would probably start by listing all the properties of O(2,R) and then maybe try to abstract a way to connect those properties to the matrix form R(α). Can anyone help me on this?

Also I would interpret the maps x → R(α)x to be a rotation of x in the plane by angle α
I would interpret x → JR(α)x to be a rotation by (α + 90°) since the J matrix rotates by 90 degrees

 

[fresh_42]

Can you define "isometries of the real plane" by formulas? What makes a mapping an isometry? Why aren't translations included, which are also distance preserving? What about reflections? Do they belong to $O(n,\mathbb{R})$, as they don't change distances either? What is $J$?

Again, some information in section 2 of the template would have been very helpful.

To write LaTex code, which is far better to read and not really difficult, you might want to look up how it is done:
https://www.physicsforums.com/help/latexhelp/
http://detexify.kirelabs.org/symbols.html

 

[MxwllsPersuasns]

Well mathematically an isometry is a distance preserving transformation so then the only thing I can really think of would be that the length of, say, x is the same as the length of Ax with A ∈ O(2,ℝ). I'm really not sure why translations aren't included in this Group to be honest. Like I said all I know about the group is the information I read on Wikipedia and given the properties of the Group I would imagine that translations would indeed be part of the group but they can't be as the question implies all elements of O(2,ℝ) have the form iof the 2D rotation Matrix. Perhaps the translations don't satisfy some of the other group axioms like closure under add/mult. or associativity or something?

The matrix J (I can't find online) but I believe has the form of (0, -1 / 1, 0) where the '/' denotes new row and I believe its acts linearly with the rotation matrix A to further rotate the vector by 90°. I believe this but I'm really not sure.

Also thank you for the link to LaTex code, I've been meaning to learn this, and I absolutely will but right now I have no time as I have too much homework due tomorrow to focus on anything but getting it done. Thanks again though, definitely going to read it over this weekend! :)

 

[fresh_42]

Translations aren't considered, because they are not linear. Linear mappings have to map $0$ to $0$ which a translation doesn't. So linearity is the key here. Wikipedia defines $O(n,\mathbb{R})=\{Q\in GL(n,\mathbb{R}\,\vert \, Q^TQ=QQ^T=1\}$. Thus reflections are allowed, because they are orthogonal with determinant $-1$. Let's take $J=\begin{bmatrix}-1&0\\0&1\end{bmatrix}$ to be the reflection along the $y-$axis and the rotations $R(\alpha)=\begin{bmatrix}\cos \alpha &-\sin \alpha\\ \sin \alpha& \cos \alpha \end{bmatrix}$. We then get $JR(\alpha)=\begin{bmatrix}-\cos \alpha &\sin \alpha\\ \sin \alpha& \cos \alpha \end{bmatrix}$.

Now the statement you want to prove is $O(n,\mathbb{R}) \stackrel{!}{=} \{R(\alpha)\, , \,JR(\alpha)\,\vert \, -\pi < \alpha \leq \pi \}$.

At this point you have to decide whether you take the definition above to define $O(n,\mathbb{R})$ or the geometric description "all planar, linear isometries". In this case, you will have to define planar, linear isometries in the first place, i.e. in terms of formulas in order to get a hand on it. Of course, you could as well show, that all three definitions define the same group, but you have to start somewhere, i.e. with one of the two (geometrical or $QQ^T=1$).