On 3/22/11, Fred lunnon <fred.lunnon@gmail.com> wrote:
Consideration of vector bases easily establishes that in projective space of dimension n-1, subspaces of dimensions k-1 and m-1 must meet in a common subspace of dimension at least k+m-n-1, when this is non-negative. ...
Just in case anybody here might be interested, I posted the following on the geometric_algebra list: principal angles were discussed here first. A recent enquiry on the math-fun list concerning a formula for the distance between two lines in Euclidean 3-space set me thinking about the more general expression for distance d between (k-1)-, (l-1)-subspaces X,Y in Euclidean (m-1)-space ["l" is "ell"]. The appropriate algebra is Cl(m-1,0,1), with vectors representing (reflections in) hyperplanes: it's time this had a name, and I propose calling it "EGA" in future. I conjecture that d^2 = ||<XºY>_{m-k-l}|| / ||<X•Y>_{k+l-2}|| where X•Y denotes Clifford product, XºY == (X*•Y*)* dual product, ||X|| == (X~)•X (scalar) magnitude, <X>_k the k-grator (grade-k part). I've verified this for all 2-space and 3-space cases, but currently have not the faintest idea how to prove it in general. It eventually dawned on me that there is a strong connection between this and the topic of principal angles, which was discussed on this list earlier (thread beginning 5th Jan 2010). There I defined a "grade expansion polynomial" whose coefficients are the magnitudes of the k-grators of X•Y, and whose roots yield the angles between the subspaces. In the case of distance, the associated root is infinite; however, the expression above shows that it is still possible to extract the actual value from the product! Fred Lunnon