distance between two linear subspaces

[wdlang]

suppose v1 and v2 are two linear subspaces of a linear subspace v

is there any measure of the distance between the two subspaces?

in two dimensional complex space, i think the distance between x and y axes is the maximum possible value. Intuitively, if two subspaces are orthogonal to each other, then their distance is of the largest possible value.

[Fredrik]

You can use a norm or an inner product to turn the vector space into a metric space (http://en.wikipedia.org/wiki/Metric_space#Definition).

d(x,y)=\|x-y\|=\sqrt{\langle x-y,x-y\rangle}

Then you can define the distance between subsets like this:

d(A,B)=\inf\{d(x,y)|x\in A,\ y\in B\}

Edit: Now I see that you said subspaces. The distance I just defined is obviously =0 if A and B are subspaces. If your vector space is an inner product space over the real numbers, you can define the angle between two vectors by

\cos\theta_{xy}=\frac{\langle x,y\rangle}{\|x\|\|y\|}

Now I suppose you could define

d(A,B)=\inf\{\theta_{xy}|x\in A, y\in B\}

but I've never seen anyone do that.

[arkajad]

Assuming you have an Euclidean space V, assuming your two subspaces are of the same dimension, then the distance is given by ||P_1-P_2||, where P_1,P_2 are orthogonal projection operators on these subspaces. This is essentially the same idea as the one mentioned by Fredrik.

[Fredrik]

Your idea should work for complex vector spaces too. Is this the norm you have in mind?

\|A\|=\sup_{\|x\|=1}\|Ax\|

[arkajad]

Can be this, or can be the trace norm |||A|||^2=Tr(A^\dag A), which may be easier to calculate.