I'm trying to prove that the flag $F_{d_1\ldots d_m}(V)=\{(S_1,\ldots,S_m)|S_1\subset S_2\subset\ldots S_m \ \ \ \ \textrm{with} \ \ \ \ \textrm{dim}S_i=d_i \ \ \ \ \textrm{and}\ \ \ 1<d_1<\ldots<d_m<n\}$ where $V$ is a complex vector space od dimension $n$ and $d_i$ fixed $\forall i=1,\ldots,m$, is a smooth manifold.
I'm using the following construction of the flag manifold: Let $GL(d_m,\mathbb{C})$ and take the subgroup of upper-triangular block matrices $U(d_m)$ where the blocks o the digonal are of sizes $d_1\times d_1, (d_2-d_1)\times(d_2-d_1),\ldots, d_m-d_{m-1}$, we will considerer the flag manifold $F_{d_1\ldots d_m}$ as the quotient space $GL(d_m,\mathbb{C})/U(d_m)$.
This construction is similar as the way we construct the Grassmanian, I tried to imitate that construction with $m=2$. I took the matrix $$ A=\left[\begin{array}[cc] &A_{d_1\times d_1}&A_{d_1\times(d_2-d_1)}\\ A_{(d_2-d_1)\times d_1}&A_{(d_2-d_1)\times(d_2-d_1)}\\ A_{(n-d_2)\times d_1}&A_{(n-d_2)\times(d_2-d_1)} \end{array}\right] $$ Where $A_{i\times j}$ is a block matrix of size $i\times j$. Then I took the matrix $$ B=\left[\begin{array}[cc] &A_{d_1\times d_1}^{-1}&B_{d_1\times(d_2-d_1)}\\ 0&A_{(d_2-d_1)\times(d_2-d_1)}^{-1}\end{array}\right] $$ Since $B$ is an upper-tringulr block matrix, we have tht $AB\sim A$, therefore $$ A\sim AB=\left[\begin{array}[cc] &I_{d_1\times d_1}&A_{d_1\times d_1}B_{d_1\times(d_2-d_1)}+A_{d_1\times(d_2-d_1)}A_{(d_2-d_1)\times(d_2-d_1)}\\ A_{(d_2-d_1)\times d_1}A_{d_1\times d_1}^{-1}&A_{(d_2-d_1)\times d_1}B_{d_1\times(d_2-d_1)}+I_{(d_2-d_1)\times(d_2-d_1)}\\ A_{(n-d_2)\times d_1}A_{d_1\times d_1}^{-1}&A_{(n-d_2)\times d_1}B_{d_1\times(d_2-d_1)}+A_{(n-d_2)\times(d_2-d_1)}A_{(d_2-d_1)\times(d_2-d_1)} \end{array}\right] $$ Now the only thing I can see from here is that $$ A\sim\left[\begin{array}[cc] &I_{d_1\times d_1}&A_{d_1\times d_1}B_{d_1\times(d_2-d_1)}+A_{d_1\times(d_2-d_1)}A_{(d_2-d_1)\times(d_2-d_1)}\\ A_{(d_2-d_1)\times d_1}A_{d_1\times d_1}^{-1}&A_{(d_2-d_1)\times d_1}B_{d_1\times(d_2-d_1)}\\ A_{(n-d_2)\times d_1}A_{d_1\times d_1}^{-1}&A_{(n-d_2)\times d_1}B_{d_1\times(d_2-d_1)} \end{array}\right]+\left[\begin{array}[cc] &0&0\\ 0&I_{(d_2-d_1)\times(d_2-d_1)}\\ 0&A_{(n-d_2)\times(d_2-d_1)}A_{(d_2-d_1)\times(d_2-d_1)}\end{array}\right] $$ If in the second column of the first matrix all the entries are zero, then I would have a parametrization of the flag. But first I don't know how to do it and second, if I find the way I would lose the element $A_{d_1\times(d_2-d_1)}$, so the maps wouldn't be injective.
Amy suggestion on how to find the charts of the flag manifolds?