= Complex Grassmannian as orthogonal projections
{title2=$P=P^*=P^2,\quad\operatorname{tr}P=k$}
The complex <Grassmannian> of $k$-planes in $\mathbb C^n$ identifies with the displayed <Hermitian> <orthogonal projection matrices>. A plane is recovered as the image of its projection. Near a plane $E$, a neighboring plane is the graph of a complex <linear map> $Z:E\to E^\perp$. Its projection is $W(W^*W)^{-1}W^*$ with $W=(I,Z)^T$. The <unitary group> acts transitively by conjugation, and its tangent directions are $[A,P]$ with $A^*=-A$.
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