There are several references in the literature to some kind of "most natural" metric on the Grassmannian manifold, often called the "geodesic distance" or the "Binet-Cauchy" distance. There are several equivalent ways to derive this metric: perhaps the simplest is to embed the (signed) Grassmannian into Euclidean space via the Plücker embedding as a projective manifold, and then the restriction of the $\ell_2$ Euclidean metric to the embedded projective manifold gives the distance between two (weighted) subspaces. A good reference on this can be found in this paper, including equivalent characterizations via "principal angles" between subspaces.
The thing is, I don't understand why this is the metric on the Grassmannian. There is another perfectly good Euclidean embedding and associated metric for the Grassmannian: the projective embedding, where each subspace is represented via the unique orthogonal projection matrix mapping to that subspace. The Frobenius norm on projection matrices then gives another perfectly good distance between subspaces.
Put simply, suppose $M$ is a full-column-rank $n \times r$ matrix, the columns of which are an orthonormal basis for some $r$-dimensional subspace of $\Bbb R^n$. Then:
- The Plücker embedding is given by the $r$'th compound matrix of $M$, notated $C_r(M)$
- The projective embedding is given by the projection matrix $M M^+$, where $M^+$ is the pseudoinverse of $M$
In both cases, you get:
- one matrix for each subspace, regardless of which basis you choose (up to sign in the first case)
- a set of polynomial equations that uniquely define the embedded variety
- an induced Euclidean distance on the result (the Frobenius norm)
But these two distances are not the same.
So what claim does the Plücker embedding have to inducing the "true" metric on the Grassmannian?
And what does this mean, that the Grassmannian corresponds to two different algebraic varieties?
Best Answer
After some searching, I found a good paper on the topic of different metrics on the Grassmannian here: "Unitarily Invariant Metrics on the Grassmann Space" by Qiu, Zhang, Li (https://doi.org/10.1137/040607605).
The short answer is that the "simplest" metric can be thought of as the $\ell_2$ norm of the vector of principal angles between the subspaces, from which these other various metrics can be derived. The Frobenius one is what this paper calls the "gap metric," whereas I believe the Plücker one is what the paper refers to as the "Hausdorff metric."
So one answer to my question is, there is no "true" metric; a different answer is that the metric involving the principle angles is an even deeper metric from which these can be derived; yet another answer is that they all give fairly similar results.