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Let´s suppose we are given two subspaces of [tex]\mathbb{R}^n[/tex] that have dimensionk, where [itex]1\leq k<n[/itex]. I think they are calledgrassmanians.

How can I compute a "distance" between two differentk-subspaces?

my attempt to a solution:

As a toy example, forn=2andk=1we can use the minimum angle between the unit-vectorsuandv:

[tex]d(u,v)=\frac{|<u,v>|}{|u||v|}[/tex]

In order to extend this to more dimensions I used the definition ofscalar productused inGeometric (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?

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# Distance between subspaces

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