Let $E$ be a finite dimensional real inner product space. I want to define the angle between two subspaces $E_1$ and $E_2$. This has a fairly obvious meaning if $E_1$ is 1-diemsnional: Take the angle between any non-zero vector in $E_1$ and its orthogonal projection onto $E_2$.
There are a number of other cases that can be treated ad-hoc, if one is a hyperplane, or the dihedral angle between planes in $R^3$.
In general, it isn't quite clear what the right definition is. I see two possibilities:
-
If $p=\dim E_1\le \dim E_2$, consider the two subspace $\lambda^p(E_1)$ and $\Lambda^p(E_2$ of $\Lambda^p(E)$ (which is also an inner product space, and proceed as above, since $\Lambda^p(E_1)$ is a line.
-
$Hom(E,E)$ is itself an inner product space with the inner product
$$
\langle A,B\rangle=trace A^\top B.
$$
Let $A_i$ be the orthogonal projection onto $E_i$ and take the angle between $A_1$ and $A_2$.
Are either of these definitions standard? Are they equivalent (I think so)? Is there another definition, perhaps more immediate?
Best Answer
There is a standard answer: Principal angles, see http://en.wikipedia.org/wiki/Principal_angles.
Let $p \ge q$ be the dimensions of the two subspaces $E_1$ and $E_2$. Then there is a unique non-increasing sequence $[c_1,c_2,...,c_q]$ with entries in $[0,1]$ (and a matching non-decreasing sequence $[s_1,s_2,...,s_q]$) such that one can have an orthonormal basis for $E$, call it $e_1,e_2,...$, in such a way that one subspace is generated by orthonormal vectors $$e_1,e_2,...,e_p$$ and the other subspace generated by orthonormal vectors $$c_1e_1+s_1e_{p+q},c_2e_2+s_2e_{p+q-1},...,c_qe_q+s_qe_{p+1}.$$ One can see this from the Singular Value Theorem. The principal angles are obviously those angles whose cosines match the $c_i$ values.
This concept captures all of the geometric invariant information relating the positioning of the two subspaces, so any well-defined definition you care to give must be a deterministic function of this sequence of principal angles.