How Can You Measure the Distance Between Two Subspaces in Higher Dimensions?

[mnb96]

Hello.
Let´s suppose we are given two subspaces of $$\mathbb{R}^n$$ that have dimension k, where $1\leq k<n$. 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:

$$d(u,v)=\frac{|<u,v>|}{|u||v|}$$

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

$$\ast : \wedge\mathbb{R}^n \times \wedge\mathbb{R}^n \rightarrow \mathbb{R}$$

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

where the (i,j) element of that matrix is $$m_{ij}=<a_i,b_j>$$.
Is that correct?

 

[hamster143]

How do you define "distance"?

 

[mnb96]

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.