I have a set of 'm' *unit* vectors, v1, v2, ... vm, with k dimensions each.

The angle between any two vectors is less than 90 degrees, by that I mean that the dot/inner product of any two vectors is positive, vi.vj >= 0 forall i,j in [1, m]

Now since all the angles between the vectors are less than 90 degrees, they can all 'fit' in one quadrant (quadrant/octant ... by quadrant i mean a region in the k-dimensional space where all the j^th co-ordinates of all the 'm' vectors have the same sign, forall j's) .

The problem is to find one such transformation matrix U such that all U*vi forall i, are in the *first quadrant*.

It suffices to give a transformation matrix to transform the set of vectors into *any* quadrant, and then we can have a set of reflections to bring them into the *first quadrant* (just change all the negative co-ordinates en-masse to positive)

Any replies will be greatly appreciated.