## A metric for Grassmannians

Hello everybody! I'm reading an article by Ricardo Mane, Hausdorff dimension is dipheomorphism. I'm having a technical problem. Sorry for my ignorance but Would you like a references where I can find explanations on how to how to equip the grassmannians manifolds with a metric.

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Given two $k$-dimensional vector subspaces $W_1$, $W_2$ of a common vector space $V$, let $d(W_1,W_2)$ be the Hausdorff distance between $W_1 \cap S$ and $W_2 \cap S$ where $S$ is the unit sphere in your vector space -- assuming your vector space has an inner product. See: en.wikipedia.org/wiki/Hausdorff_distance – Ryan Budney Dec 3 2010 at 0:51
Another way is to view the Grassmannian as a homogeneous space, i.e. as a quotient of a Lie group by a compact subgroup. Homogeneous spaces inherit metrics as quotient spaces, since compact Lie groups have invariant metrics. – Ryan Budney Dec 3 2010 at 0:53

I found it surprisingly difficult to find a reference for this when I was studying Mane's papers on multiplicative ergodic theorems. My hypothesis was that people working with the Grassmannian in other areas are happy with the fact that the Grassmannian is metrisable for abstract topological reasons, and don't actually care very much about a precise metric, but I might be wrong about this... in my answer I'm going to assume that we're considering a finite-dimensional space equipped with an inner product structure.

If you are interested in precise metrics on the Grassmannian, the most popular definition of which I am aware is this one: $$d(V,W):=\max\left\{\sup_{w \in W, \|w\|=1}\inf \{\|v-w\| \colon v \in V \},\sup_{v \in V, \|v\|=1}\inf \{\|v-w\| \colon w \in W\}\right\}$$ This is I think not quite the same as the one suggested by Ryan Budney, but produces the same topology. This one seems to be the most popular definition for people working in multiplicative ergodic theory (it is in Barreira and Pesin's book, for example).

There are some equivalent ways of describing this metric which seem to be less well-known. If we know a priori that $V$ and $W$ have the same dimension, then the maximum in the expression above is always attained by both expressions simultaneously! Hence if we fix a dimension $r$, then the expression $$d(V,W):=\sup_{v \in V,\|v\|=1}\inf\left\{\|v-w\|\colon w \in W\right\}$$ is actually a metric for the component of the Grassmannian which consists of all $r$-dimensional subspaces. This does not seem to be very well-known; I actually discovered this by reading Kato's book on perturbation theory, which isn't exactly the first place I'd go to to find out about Grassmannian manifolds...

Another way to put a metric on the Grassmannian is as follows. We can identify a subspace $U$ with the unique linear operator of orthogonal projection onto that subspace, and take the metric given by setting the distance between two subspaces to be the operator norm distance between the orthogonal projection operators corresponding to those subspaces. I personally like this approach a great deal, because I think it makes it very obvious that the Grassmannian is compact (well, obvious if you're a functional analyst!). This metric is also, rather pleasantly I think, exactly identical to the first metric I defined above. You can find a proof that the two things are the same in the book on Hilbert spaces by Akhiezer and Glazman.

There's a short discussion on this topic in my paper "A rapidly-converging lower bound for the joint spectral radius via multiplicative ergodic theory", which is basically the result of a gentle argument between myself and the referee over how the metric on the Grassmannian should be defined!

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Many thanks to David Speyer for the displayed equation fix! – Ian Morris Dec 6 2010 at 16:40
I took the liberty of cleaning up your LaTeX. The issue is that you need to surround some of your equations with backticks (the symbol to the left of the number 1, on an american QWERTY keyboard). Otherwise, some of the backslashes and underscores will get scrambled by the markdown interpreter before LaTeX gets to see them. – David Speyer Dec 6 2010 at 16:42
How are any of the sets above, which are being sup'ed over, bounded? In each expression one vector can have arbitrary magnitude, no? I don't understand. – AndrewLMarshall Dec 6 2010 at 18:17
I thought of the case inf was meant, but then k-planes with non-trivial intersections are distance 0 from each other. Shouldn't it be, maybe, inf over w, sup over v, ||v||=1, similarly to how the Hausdorff metric is defined? – AndrewLMarshall Dec 6 2010 at 19:00
Do I miss something if I say that when $V\cap W$ is nonzero, your $d(V,W)$ vanishes ? – BS Dec 6 2010 at 19:03
The $k$-dimensional Grassmannian over a vector space $V$ naturally embeds into the projectivization of the $k$-th exterior power of $V$. If you have a Euclidean structure on $V$, then it extends to its exterior products as well. Then a metric on the projectivization is provided, say, by the angle between the corresponding directions, or by sine of that angle.