Orthogonal transformation and mirror transformation

  • #1
How to prove any orthogonal transformation can be represented by the product of many mirror transformations, please?What's the intuitive meaning of this proposition?
Thank you.
 

Answers and Replies

  • #2
Simon Bridge
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You prove it by comparing the definitions, and writing an expression relating the two. You know, how you normally go about proving something in mathematics.
Start by writing out the mathematical expression for a mirror transformation, and also for an orthogonal transformation. What are they, how do they work? How general do you need the proof?

There does not need to be an intuitive idea embodied in a proposition. However, if there is one, it should emerge from the definitions of the terms.
 
  • #3
You prove it by comparing the definitions, and writing an expression relating the two. You know, how you normally go about proving something in mathematics.
Start by writing out the mathematical expression for a mirror transformation, and also for an orthogonal transformation. What are they, how do they work? How general do you need the proof?

There does not need to be an intuitive idea embodied in a proposition. However, if there is one, it should emerge from the definitions of the terms.
Def:AL(V,V) V is Euclidean space ,dim V=n
A is mirror transformation ⇔ A ( α ) =α - 2 ( η, α ) η ( η∈V and ||η||=1) (∀α ∈ V)
Def:AL(V,V) V is Euclidean space ,dim V=n
A is orthogonal transformation ⇔ ( A(α) , A(β) )=(α,β). (∀α,β∈V)

Question:
Prove: If A is an orthogonal transformation over V (Euclidean space)
⇒∀α∈V A(α)=B1B2````BK (α) ( Bi is a mirror transformation over V , i=1,2······k)

(Note : When n=1 and n=2 it can be proved, then I am confused about higher dimension )
 
  • #4
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When n=1 and n=2 it can be proved, then I am confused about higher dimension
If you have it for ##n=2## have you considered to do it by induction? It would help to see how you've done it.
 
  • #5
WWGD
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Do you also have these matrices by a different name? I could not find anything on mirror matrices. This seems vto day that ortho matrices are generated by these mirror matrices
 
  • #6
Simon Bridge
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As above ... and it can help to pick a specific orthogonal transformation and see what happens, so you get a feel for how the transformations work.
The "intuitive" principle you are exploiting is that the reflection of a reflection is the right way around.
If you prefer, you can put a 1:1 scale image anywhere, in any orientation you like, by use of strategically placed mirrors.
 
  • #7
WWGD
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As above ... and it can help to pick a specific orthogonal transformation and see what happens, so you get a feel for how the transformations work.
The "intuitive" principle you are exploiting is that the reflection of a reflection is the right way around.
If you prefer, you can put a 1:1 scale image anywhere, in any orientation you like, by use of strategically placed mirrors.
Ah I see, so the mirror transformation s are reflections? I think I remember orthogonal transformation s we're generated by shear maps. Is that what these mirror maps are?
 
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  • #8
Simon Bridge
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You can check to see if what you think of as a reflection fits the definition of a "mirror transformation" given above.
 
  • #9
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Hi Leo-physics,
if you know how to prove this for a rotation in ##\mathbb R^2##, then you can get the desired statement from results in s.5 of Chapter 6 of "Linear Algebra Done Wrong", see Theorems 5.1, 5.2 there.
 

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