Non-Rotation Matrix Split: Hello

In summary, the conversation discusses the possibility of splitting a general matrix M into the product of a rotation matrix R and "something else" represented by S, such that M = RS or M = R + S. The question also mentions the difference between 2D and 3D cases and the use of SE(2) and SE(3) matrices for separating rotations from translations. The polar decomposition is suggested as a possible solution.
  • #1
Trying2Learn
373
57
Hello

This could very well be an idiotic question, but here goes...

Consider a general matrix M
Consider a rotation matrix R (member of SO(2) or SO(3))

Is it possible to split M into the product of a rotation matrix R and "something else," say, S?
Such that: M = RS or the sum M = R + S

What would that something else, S, be?

Would 2D and 3D be different?

Oddly, despite the stupidity (I fear it may be an idiotic question), I am aware of SE(2) (3x3 matrix) and SE(3) (4 x4 matrix) and their roles in separating rotations from translations. So I would rather not go down that path.
 
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  • #2
The polar decomposition is something like this. In its definition, a unitary matrix is used in place of a rotation matrix, to allow complex-valued matrices.
 

1. What is a non-rotation matrix split?

A non-rotation matrix split is a mathematical process used to decompose a matrix into two parts, one representing a rotation and the other representing a non-rotation. This can be useful in certain applications, such as computer graphics and image processing.

2. How is a non-rotation matrix split different from a regular matrix split?

A regular matrix split decomposes a matrix into two parts that are both rotations. In contrast, a non-rotation matrix split separates a matrix into a rotation and a non-rotation, which can be useful in cases where a rotation is not desired.

3. What are the applications of non-rotation matrix split?

Non-rotation matrix split can be used in computer graphics to separate the rotation and scaling components of a transformation matrix. It can also be used in image processing to separate the orientation and scale of an image from other transformations.

4. Can any matrix be split into a rotation and non-rotation?

No, not all matrices can be split into a rotation and non-rotation. This decomposition is only possible for matrices that are square and have a determinant of 1 or -1.

5. What are the advantages of using a non-rotation matrix split?

One advantage of using a non-rotation matrix split is that it can simplify complex transformations into separate rotation and non-rotation components, making it easier to understand and manipulate. It can also be useful in certain applications, such as robotics and computer vision, where the rotation and non-rotation components may need to be handled separately.

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