# Parameterization Of Grassmannian

Is there an explicit formula to parameterize the Grassmannian $P$ (which is a $(a+b)\times (a+b)$ dimensional matrix) $$P\in \frac{U(a+b)}{U(a)\times U(b)}$$ by $a\times b$ complex independent parameters?

(One may normalize $P$ such that $P^{\dagger}=P, P^2=I_{N}$ where $I_N$ is the $N\times N$ identity matrix. )

• The wikipedia article en.wikipedia.org/wiki/Grassmannian explains this. Sep 8, 2018 at 2:38
• @AlanMuniz Thanks! I don't see the wiki page contains a square matrix which has extra unitary structure. Could you point it out? I appreciate your help. Sep 8, 2018 at 20:13
• It is in the section "The Grassmannian as a homogeneous space". The section above it explains how to find the coordinate charts. Sep 8, 2018 at 20:28

There is a nice local parametrization of the Grassmannian around a fixed subspace. Fix an $a$-dimensional subspace $V\subset\mathbb C^{a+b}$ and consider its orthocomplement $V^\perp$. The parametrization covers $U:=\{W:W\cap V^\perp=\{0\}\}$, which is an open subset in the Grassmannian. Any subspace in this set is the graph of a unique linear map $V\to V^\perp$ (with $V$ itself corresponding to the zero map). To see this, consider the two projections corresponding to $V\oplus V^\perp\cong\mathbb C^n$. By definition of $U$, for any $W\in U$, the restriction of the first projection to $W$ is injective and thus a linear isomoprhism. The map corresponding to $W$ then is the composition of the inverse of the first projection with the restriction of the second projection.
To get an explicit parametrizations, chose orthonormal bases $\{v_1,\dots,v_a\}$ for $V$ and $\{\tilde v_1,\dots,\tilde v_b\}$ for $V^\perp$. Then for a subspace $W\in U$, and each $i=1,\dots,a$, there are unique complex numbers $c_{ij}$ for $j=1,\dots,b$ such that $v_i+\sum_{j=1}^bc_{ij}\tilde v_j$ lies in $W$. Conversely, you associate to a matrix $(c_{ij})$ the subspace spanned by the vectors $v_i+\sum_{j=1}^bc_{ij}\tilde v_j$ for $i=1,\dots,a$, which are linearly independent by construction.