Non-Rotation Matrix Split: Hello

[Trying2Learn]

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.

 

[hilbert2]

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.