Proving Reflections in R^n are Symmetric Matrices

[Pearce_09]

done

Hi there,
I am have trouble with a proof. I have some steps done, but I am not sure if I am aproaching this correctly.
the question is:

Show that the matrix of any reflection in R^n is a symmetric matrix.

I know that F(x) = Ax + b is an isometry
where A is an Orthoganol matrix , and vector b is in R^n
this implies that A^t A = I (identity matrix) iff A^-1 = A^t

and if f is a reflection then f(f(x)) = x
then f(f(x)) = A(Ax + b) + b = x
_________
i need to somehow prove that A = A^t which then means the matrix is symmetric

thank you for you time and help
regards,
adam

 

[AKG]

Not every reflection in Rⁿ can be represented by a matrix. Now, do you mean that (1) every reflection in an n-1 dimensional hyperplane passing through the origin in Rⁿ is a symmetric matrix, or do you mean (2) the matrix part of every reflection in Rⁿ is symmetric? If f is a reflection in an n-1 dimensional hyperplane L, then f(x) = Ax + b, where A is the matrix corresponding to a reflection in the n-1 dimensional hyperplane passing through the origin, parallel to the hyperplane L, and if d is the shortest vector from 0 to L, then b = 2d. You know that A is orthogonal, so you know that A^-1 = A^T. It remains to show that A^-1 = A. But this is true since:

A(Ax + b) + b = x
A²x + Ab + b = x
(A² - I)x = -(Ab + b)

Now you could use the fact that Ab = -b, so the right side is:

-(Ab + b) = -(-b + b) = -0 = 0

for all x, hence A² - I = 0, hence A² = I, and thus A = A^-1. But suppose you didn't know what Ab was. Regardless, you still know that the right side is:

-(Ab + b) for all x

but this is a constant, so you have (A² - I)x = y, where y is a constant vector, and this is true for all x. If you plug in x = 0, then you get 0 = y. In other words, the only linear function that is constant is the 0 function. So without knowing explicitly what Ab is, you can deduce still that the right side must be 0 (and so, if you wanted, could then deduce that Ab = -b), and since the right side is 0 for all x, A² - I = 0, and the desired result follows.

 

[Pearce_09]

Oh wow, that really clears things up. thank you very much akg..