The Grassmannian As A Real Affine Algebraic Variety
Let Gr(r, Rn) denote the Grassmannian of r-dimensional subspaces of Rn. Let M(n, R) denote the space of real n-by-n matrices. Consider the set of matrices A(r, n) ⊂ M(n, R) defined by X ∈ A (r, n) if and only if the three conditions are satisfied:
- (i.e.: it is a projection operator)
- (it is symmetric)
- (its trace is r)
A(r, n) and Gr(r, Rn) are homeomorphic, with a correspondence established by sending X ∈ A(r, n) to the column space of X.
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