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I know how to build a rotation matrix around the x, y or z axis, but i'm not sure how to do it around an arbitrary vector. For example, in R3, how should i build a rotation matrix with axis (111)? I read i should pick an orthonormal basis but i dont get why it cant be any R3 basis.

Also, i dont understand if a rotation matrix should always be an orthogonal (or hermitian) matrix. I mean, what if i define a linear transformation R as a rotation around the z axis, and apply a change of basis:

[R]b = Ceb.[R]e.Cbe

with R being the rotation matrix and Cbe the change of basis matrix from basis e to b. The resultant matrix wont be orthogonal but still is a matrix defining a rotation, isn't it?

Thanks!

Federico A
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  • https://en.wikipedia.org/wiki/Rodrigues%27_rotation_formula – amd Feb 11 '20 at 06:39
  • @Jose I don't think that this is a duplicate; there are some unique aspects to the question. In particular, why we need an orthonormal basis, and why rotation matrices are always orthogonal – Ben Grossmann Feb 11 '20 at 06:47
  • Here is a post that asks for a rotation about the same axis, and here is a post that asks why the new basis needs to be orthonormal. – Ben Grossmann Feb 11 '20 at 06:53
  • A rotation is a rigid motion, hence it must preserve distances and angles, or equivalently, it must preserve inner products. Thus for any vectors $x,y$, a rotation matrix $R$ must satisfy $\langle x,y\rangle = \langle Rx, Ry\rangle$, which is equivalent to $x^T y = x^T R^T R y$. Choosing $x = e_i$ and $y = e_j$ (the canonical $i$'th and $j$'th basis vectors) shows that the columns of $R$ must be orthonormal. –  Feb 11 '20 at 07:04

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