Understanding Rotation Matrices for nxn Orders

In summary, the rotation matrix in Rn can be represented as a matrix with determinant 1, and in R2 and R3, it is composed of perpendicular unit vectors rotated theta degrees from the original vectors. In higher dimensions, it can be expressed as a combination of sequential plane rotations or reflections. The minimum number of sequential plane rotations needed to represent a general rotation in Rn is n-1 for odd n and n-2 for even n.
  • #1
MathematicalPhysicist
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I know how the rotation matrix looks like in the 2x2 and 3x3 orders, but how does it look in general?

thanks in advance.
 
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  • #2
I know that, in 2 dimensions, a rotation about (0,0) by angle [itex]\theta[/itex] can be represented as a matrix by
[tex]\left[\begin{array}{cc}cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta)\end{array}\right][/tex]
but what are you saying a rotation in 3 dimensions looks like?

Any matrix with determinant 1 can represent a rotation.
 
  • #3
Think of a rotation as taking the x,y,z,w,... axes, and rotating them around to a new set of axes x2,y2,z2,w2,... If you take the unit vectors in the representing of the new axes, and put them in the columns of a matrix in their respective order, then that matrix you get will be the rotation matrix.

In R2, you may notice that the vector (cos(theta),sin(theta)) and (-sin(theta),cos(theta)) are perpendicular unit vectors rotated theta degrees from the original (1,0) and (0,1) vectors.
 
  • #4
If you are looking for a R^N rotation in a plane, rotating the components of a vector that lies in the plane and leaving the "perpendicular" components (ie: components in all normal directions) untouched, then this can be expressed with the following geometric-algebra/clifford-product expansion:

[tex]
R(x) = e^{-i\theta/2} x e^{i\theta/2}
= x \cos^2(\theta/2) - i x i \sin^2(\theta/2) + 2 \cos(\theta/2) \sin(\theta/2) x \cdot i
[/tex]

Here the unit bivector i is formed by the product of two non-colinear vectors n_k in the plane:

[tex]
i=\frac{n_1 \wedge n_2}{\lvert n_1 \wedge n_2 \rvert}.
[/tex]

It's a bit messy, but you can calculate the rotation matrix from this if you really wanted to (with an orthonormal basis basis e_i and calculation of R(e_i) \cdot e_j gives the matrix).
 
  • #5
another way to look at this is, just like so:

[tex]
R(x) = x_{\perp} + x_{\parallel}\left( cos\theta + i\sin\theta\right)
[/tex]

where [itex]x_{\perp}[/itex] is the component not in the plane of rotation, [itex]x_{\parallel}[/itex] is the component in the plane of rotation, and [itex]x_{\parallel} i [/itex] is a 90 degree rotation in the desired sense of the component of the vector in the plane. Then given any way you are comfortable with (ie: projection and rejection matrixes P, and (I-P)) for finding the components of the vector in and out of the plane, you can use that to compute the rotation matrix.
 
  • #6
i think a rotation is an orthogonal matrix with determinant one?

so there is some basis in which the matrix looks like a bunch of 1's on the diagonal, and then a bunch of 2 by 2 matrices representing plane rotations.
 
  • #7
Yes, thanks mathwonk I also got this answer after solving a question from Gullimin and Pollack:" if k is odd then f:S^k->S^k f(x)=-x is homotpic to the identity function".

The homotopy is a sort of rotation matrix times x, where the argument is (pi*t), so if k is odd we can find a rotation matrix (k+1)x(k+1) (k+1 is even) which consists of orthogonal matrices on the diagonal representing plane rotations.

P.S
You got to love geometry! (-:
 
  • #8
So that raises an interesting question - what is the minimum number of sequential plane rotations required to represent a general rotation in Rn?

In R2, its just 1. In R3 its 2 since you rotate xhat to xhat', and then roll around xhat'. What about higher dimensions?
 
  • #9
maze said:
So that raises an interesting question - what is the minimum number of sequential plane rotations required to represent a general rotation in Rn?

In R2, its just 1. In R3 its 2 since you rotate xhat to xhat', and then roll around xhat'. What about higher dimensions?

Every plane rotation can be factored into two reflections. It might be easier to think in terms of reflections.

There's a really elegant derivation of this (i.e., the number of plane rotations needed to give a general rotation) in The Road to Reality using Clifford algebras, but I forget it.
 
  • #10
well if we have R^n where n is odd, then there must be 1 in the diagonal of the matrix, while in an even n we can handle without it (if we want).
 

1. What is a rotation matrix?

A rotation matrix is a mathematical tool used to describe the rotation of an object in three-dimensional space. It is a square matrix that contains a combination of cosine and sine functions, and is used to transform coordinates from one coordinate system to another.

2. How do rotation matrices work?

Rotation matrices work by multiplying a vector representing a point in space by the rotation matrix. This results in a new vector that represents the rotated point. The rotation matrix itself is made up of a combination of cosine and sine functions that determine the amount and direction of rotation.

3. What is the order of a rotation matrix?

The order of a rotation matrix refers to the size of the matrix (e.g. 2x2, 3x3, etc.). This determines the dimension of the space in which the rotation is taking place. For example, a 2x2 rotation matrix would be used for two-dimensional rotations, while a 3x3 rotation matrix would be used for three-dimensional rotations.

4. How do I create a rotation matrix for a specific angle?

To create a rotation matrix for a specific angle, you can use the formula:

R = [cos(theta) -sin(theta)
sin(theta) cos(theta)]

Where theta is the desired angle of rotation. This will create a 2x2 rotation matrix. For higher order rotation matrices, the process is more complex and may involve using trigonometric identities.

5. What are some practical applications of rotation matrices?

Rotation matrices have many practical applications, including computer graphics, robotics, and physics. They can be used to rotate objects in computer animations, determine the position of a robot arm, or calculate the angular momentum of an object in motion. They are also useful for solving problems in vector calculus and mechanics.

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