Minimizing a quadratic function w.r.t. a rotation matrix
I have an objective function of the following quadratic form:
where is an NxN matrix (properties described below), and are Nx1 vectors, and is an NxN rotation matrix (including reflections), s.t. (identity). I would like to minimize , given that constraint.
I would like to know how this might most efficiently be achieved in the following cases:
1. , being a scalar.
2. , being an Nx1 vector.
3. is a positive semi-definite (symmetric) matrix.
It is worth noting that if there is a specific or simpler solution for N = 3, then I'd be interested in that, though the general solution is of primary interest.
Can you help?