In an n-dimensional Euclidean space, what is the minimum number of rotations required to transform a given orthogonal coordinate system into an arbitrary orthogonal coordinate system? How can this be expressed mathematically, particularly using the language of linear algebra?
A Given rotation is a rotation in a plane spanned by two coordinate axes, leaving all other coordinates unchanged. For example, in 3D space, a rotation in the xy-plane affects only the x and y coordinates, leaving z unchanged.
The minimal number of rotations required to align two arbitrary orthonormal bases in n-dimensional space is equal to the number of degrees of freedom in an orthogonal matrix with determinant +1, which is
n(n-1)/2.
This is because each rotation can be thought of as eliminating one degree of freedom in the misalignment between the two bases.
Indeed, Ayman, and the 3D space ('our' space!) is the only one where the number of rotation angles n(n-1)/2 is equal to n. This allows one to visualize the total rotation in 3D as a rotation about a unique vector with 3 componets, acting as 'the' axis of the total rotation. No such vector can exist in 2D, 4D, or more.
One thought - presumably you cannot do this transformation between right handed and left handed systems?
You're correct, James. The orthonormal matrices herein are all supposed to have +1 as their determinant. So, they are also supposed to be real , not complex.
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