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Something about rotation matrices

22/04/2011

Question: A matrix A is called orthogonal iff A^TA=I. It is further called rotation matrix if A^TA=I, \det A=1. These are well known. But how to bridge the algebraic definite of rotation matrices and their geometric role?

I think the most essential geometric role of a rotation matrix is to rotate every vector in R^n the same rotation angle around the same rotation axis. Therefore, geometrically, a rotation matrix is determined by a rotation angle and and rotation axis. Now the question is actually how to understand the relationship of the following algebraic and geometric definition:

Algebraic: A^TA=I, \det A=1
Geometric: rotation angle \theta and rotation axis v

In other words, prove

1. Given A^TA=I, \det A=1, find the rotation angle and axis.
2. Given rotation angle \theta and rotation axis v, prove the corresponding matrix satisfy: A^TA=I, \det A=1

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