Originally published at 狗和留美者不得入内. You can comment here or there.
We all know of real projective space . It is in fact a special space of the Grassmannian manifold, which denoted
, is the set of
-dimensional subspaces of
. Such can be represented via the ranges of the
matrices of rank
. On application of that operator we can apply any
and the range will stay the same. Partitioning by range, we introduce the equivalence relation
by
if there exists
such that
. This Grassmannian can be identified with
.
Now we find the charts of it. There must be a minor 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
columns is one of such, for the convenience of writing
, where the
is
. We get
.
Thus the degrees of freedom are given by the matrix on the right, so
. 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
would alter the identity matrix on the left.
I’ll leave it to the reader to run this on the real projective case, where .