I have an $N$-dimensional space $\Bbb{R}^N$. I have a basis $B$ in $\Bbb{R}^N$, made up of $N$ $N$-vectors of unit length. I have another basis $C$ in the same space. I want to measure how "different" $B$ and $C$ are in some way. I put that in quotes because I'm not sure how that should be defined, so I'm open to ideas.
I've been trying to use the amount of rotation. I can picture it in 2 or 3 dimensions, as a single rotation on a unique plane of rotation. When $N>3$, there are $\lfloor{N/2}\rfloor$ planes of rotation and their associated angles of rotation required to describe the rotation, and my ability to visualize is gone.
If I could take my bases, and calculate their planes and angles of rotation, I would probably be happy. I could somehow combine the $\lfloor{N/2}\rfloor$ angles of rotation to arrive at some summary number of different-ness (details TBD). But:
a) I haven't been able to find a method to determine the planes/angles of rotation from actual values in many dimensions, just that it is theoretically possible.
b) There might be a better way to measure "different".
If it helps, my bases are eigenvectors of covariances of two noisy samples of $N$-dimensional data. My application has $N$ between 5 and 30, give or take.
To show I'm trying, so far I've found the pairwise angles between all the vectors in the two bases. I can then use the Hungarian/Munkres algorithm to find the best pairings of vectors in the two bases such that the sum of the angles between the pairs is minimized. I consider this to have ordered the vectors so that the $i$th vector in $B$ is best matched to the $i$th vector in $C$ (I believe these are called a frames, rather than bases, once ordered). I feel this is progress. But it leaves me with $N$ angles, when I know that this is theoretically representable by $\lfloor{N/2}\rfloor$ angles. There must be redundant information in there. Or I might be totally on the wrong track.
I've gotten this far using wikipedia. My math training is zilch. Please be gentle.