Does an Improper Orthogonal Matrix Always Have a Determinant of -1?

In summary, the conversation discusses the question of whether a 3x3 improper orthogonal matrix Q satisfies Q^2 = I. The participants initially try to prove it true but eventually realize it is not true for dimensions greater than 2. They also discuss how improper matrices indicate reflections and give hints on finding a counterexample. In the end, a counterexample is found and it is stated that improper matrices are generated by rotations and reflections.
  • #1
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The question is (true or false) if Q is an improper 3 x 3 orthogonal matrix then Q^2 = I.

The way I have approached it so far has been a brute force method. I'm not really sure if this will be true or false, and I have a feeling it is false, but I can't construct a good counter-example. So, I have been trying to prove it is true, which is becoming tedious and lengthy as I go through the inner products.

I started on another more algebraic approach too. I know that [tex]Q^T Q = I[/tex], so if QQ = I, then [tex] Q = Q^{-1} = Q^T[/tex]. Also det(Q) = -1, since it is improper. From here I am not quite sure either.
 
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  • #2
Well, it's true in even dimensions. I don't know what more to say without giving it away.

Edit: Ok, I'll tell you this: stop trying to prove it.
 
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  • #3
Yes, in an earlier question I found it was true for a 2x2, and I can see how that would extend for any even dimensioned matrix.

My idea in trying to prove the 3x3 case true was to at some point find where the preposition becomes false. Usually it is pretty easy, for me at least, to find counter examples to matrices, but with the orthogonal matrix I am having trouble thinking of a counter example since the orthogonal matrix is so limiting.
 
  • #4
You build one up out of lower dimensional matrices.

Edit: Actually, I don't think it's true for any dimension greater than 2. Sorry.
 
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  • #5
How did you prove it for 2 dimensions? That might suggest what you need to find a counterexample in 3 dimensions.
 
  • #6
It's obviously not true: think geometrically.
 
  • #7
matt grime said:
It's obviously not true: think geometrically.

What do you mean? If you have a set of 3 orthogonal vectors that form an orthogonal matrix, which must mean that the vectors themselves are orthonormal, and if you find the inner product of all the vectors then they will not necessarily still be orthonormal in lR^3?
 
  • #8
Umm. What? Orthogonal matrices are generated by rotations and reflections. In R^2, any improper matrix is a reflection. The question asks is this true in any dimension, and the answer is clearly no. So, prove that the composition of a reflection in a plane and a rotation is or is not in general a reflection in another plane.

(I think we all agree it isn't - so just find a counter example. Status X's hint gives you the answer easily - treat things as block matrices, and remember that proper orthogonal matrices (rotations) are not, in general, self inverse
 
  • #9
Okay, I think I was making it a lot tougher than it should have been. First counter example I tried worked.

I didn't realize that an improper matrix indicated a reflection though. Your geometrical response makes sense now.
 

FAQ: Does an Improper Orthogonal Matrix Always Have a Determinant of -1?

1. What is an improper orthogonal matrix?

An improper orthogonal matrix is a square matrix with real elements that has a determinant of -1, meaning it is not a proper orthogonal matrix. This means that it is not a rotation matrix, but rather a reflection or a combination of a rotation and reflection.

2. How is an improper orthogonal matrix different from a proper orthogonal matrix?

A proper orthogonal matrix has a determinant of 1 and represents a pure rotation in n-dimensional space, while an improper orthogonal matrix has a determinant of -1 and represents a reflection or a combination of a rotation and reflection.

3. What are some applications of improper orthogonal matrices?

Improper orthogonal matrices are commonly used in computer graphics and computer vision, where they are used to create reflections and mirror images. They are also used in physics, particularly in quantum mechanics, to describe spin states of particles.

4. How do you determine if a matrix is an improper orthogonal matrix?

To determine if a matrix is an improper orthogonal matrix, you can calculate its determinant. If the determinant is -1, then the matrix is an improper orthogonal matrix. Alternatively, you can also check if the matrix is equal to its own inverse, as improper orthogonal matrices are their own inverses.

5. Can two improper orthogonal matrices multiply to give a proper orthogonal matrix?

No, two improper orthogonal matrices cannot multiply to give a proper orthogonal matrix. This is because the determinant of the resulting matrix would be -1, which is not equal to 1, the determinant of a proper orthogonal matrix.

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