Grassmannian manifold

We all know of real projective space \mathbb{R}P^n. It is in fact a special space of the Grassmannian manifold, which denoted G_{k,n}(\mathbb{R}), is the set of k-dimensional subspaces of \mathbb{R}^n. Such can be represented via the ranges of the k \times n matrices of rank k, k \leq n. On application of that operator we can apply any g \in GL(k, \mathbb{R}) and the range will stay the same. Partitioning by range, we introduce the equivalence relation \sim by \bar{A} \sim A if there exists g \in GL(k, \mathbb{R}) such that \bar{A} = gA. This Grassmannian can be identified with M_{k,n}(\mathbb{R}) / GL(k, \mathbb{R}).

Now we find the charts of it. There must be a minor k \times k with nonzero determinant. We can assume without loss of generality (as swapping columns changes not the range) that the first minor made of the first k columns is one of such, for the convenience of writing A = (A_1, \tilde{A_1}), where the \tilde{A_1} is k \times (n-k). We get

A_1^{-1}A = (I_k, A_1^{-1}\tilde{A_1}).

Thus the degrees of freedom are given by the k \times (n-k) matrix on the right, so k(n-k). If that submatrix is not the same between two full matrices reduced via inverting by minor, they cannot be the same as an application of any non identity element in GL(k, \mathbb{R}) would alter the identity matrix on the left.

I’ll leave it to the reader to run this on the real projective case, where k = 1, n = n+1.

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