What is the alternative representation of SO(2) and its significance?

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In summary, there are two common representations of an element of SO(2): the usual representation using the matrix with cosine and sine values, and another representation using a matrix with sine and cosine values. Both matrices have det=1 and are equal to their own transpose, making them valid representations of SO(2). However, the second representation is not commonly used as it does not have a clear and intuitive interpretation.
  • #1
spaceofwaste
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The usual representation I see of an element of SO(2) is:

[tex] \left( \begin{array}{ c c } cos(\theta) & sin(\theta) \\ -sin(\theta) & cos(\theta) \end{array} \right) [/tex]

and it is easy to show that if you make a passive rotation of a cartesian frame by [tex]\theta[/tex] then this matrix will take the comps of an arbitrary vec to those in the new rotated frame.

However this matrix:

[tex] \left( \begin{array}{ c c } sin(\theta) & cos(\theta) \\ -cos(\theta) & sin(\theta) \end{array} \right) [/tex]

is also a valid representation of SO(2), since it has det=1, and transpose equal to inverse. I have played about with a few drawings but just don't see what this actually represents.
 
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  • #2
In your second matrix, theta is the angle between the original y-axis and the transformed x axis, with positive theta denoting a clockwise rotation. Alternatively, it is the angle from the transformed x-axis to the original y axis, with positive theta denoting a counterclockwise rotation. Neither interpretation is particularly useful or intuitive, which is why it isn't used.
 

1. What is SO(2)?

SO(2) stands for special orthogonal group in two dimensions. It is a mathematical group that represents rotations in a two-dimensional space. In simpler terms, it is a set of all possible rotations that can be performed on a two-dimensional shape without changing its size or shape.

2. How is SO(2) represented?

SO(2) can be represented in various ways, including matrices, complex numbers, and quaternions. The most common representation is through 2x2 rotation matrices, where each element in the matrix represents a rotation in the x-y plane.

3. What is the significance of SO(2) in mathematics?

SO(2) has significant applications in various mathematical fields, such as geometry, linear algebra, and differential equations. It is also used in computer graphics and robotics to represent and manipulate two-dimensional objects.

4. How is SO(2) related to other mathematical concepts?

SO(2) is a subgroup of the special orthogonal group SO(n), which represents rotations in n-dimensional space. It is also a subgroup of the general linear group GL(2), which represents all invertible linear transformations in two dimensions.

5. What are some real-world applications of SO(2)?

SO(2) has practical applications in various fields, including physics, engineering, and computer science. For example, it is used in navigation systems to track the orientation of objects, in robotics for motion planning and control, and in image processing for object recognition and tracking.

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