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?

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