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Distance between subspaces

  1. Apr 11, 2010 #1
    Let´s suppose we are given two subspaces of [tex]\mathbb{R}^n[/tex] that have dimension k, where [itex]1\leq k<n[/itex]. I think they are called grassmanians.
    How can I compute a "distance" between two different k-subspaces?

    my attempt to a solution:
    As a toy example, for n=2 and k=1 we can use the minimum angle between the unit-vectors u and v:


    In order to extend this to more dimensions I used the definition of scalar product used in Geometric (Clifford) Algebra, which is:

    [tex]\ast : \wedge\mathbb{R}^n \times \wedge\mathbb{R}^n \rightarrow \mathbb{R}[/tex]

    [tex]A\ast B=(a_1\wedge\ldots\wedge a_k)\ast(b_1\wedge\ldots\wedge b_m)=det[m_{ij}][/tex] ; for [tex]k=m[/tex]
    [tex]A\ast B=0[/tex] ; for [tex]k\neq m[/tex]

    where the (i,j) element of that matrix is [tex]m_{ij}=<a_i,b_j>[/tex].
    Is that correct?
  2. jcsd
  3. Apr 11, 2010 #2
    How do you define "distance"?
  4. Apr 12, 2010 #3
    I thought of quantifying the "distance" between the subspaces with the minimum angle between them (see the toy-example in 2-dimensions in my first post).

    The scalar product (and contraction) introduced in Geometric (Clifford) Algebra is used to compute the angle between subspaces of same (or different) dimensions.
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