I am trying to prove that the following is a diffeomorphism $$ f:D \times O(n) \rightarrow \text{GL}(n,\mathbb{R}),~~ A,C \mapsto A\cdot C$$ where $D$ is the set of real upper triangular $(n\times n)$-matrices with positive elements in the diagonal. I know this the QR decomposition under the hood, and this map is easily seen to be a differentiable one between manifolds.
In order to prove it is a diffeomorphism, it suffices to show then that it is bijective and that it is a regular map, i.e. the differential $$f_{*(A,C)}:T_A D\times T_C O(n)\rightarrow T_{AC}\text{GL}(n,\mathbb{R})$$ is a linear isomorphism. For the bijectivity part this is simply the QR decomposition (which is based in Gram-Schmidt orthogonalization process). However, I have been unsuccessful at proving that $f_{*(A,C)}$ is an isomorphism.
By counting dimensions I can either prove it is injective or surjective. Let us try the latter: let $M \in T_{AC}\text{GL}(n,\mathbb{R})\simeq \text{End}(\mathbb{R}^n)$. I need to find $X\in T_A D \simeq D_0 :=$ upper triangular matrices and $Y\in T_C O(n)$ such that
$$ X C + A Y = M$$
Here, $X$ is an upper triangular matrix and $Y$ obeys $Y^t C + C^t Y = 0$ (since $Y\in T_C O(n)$). What property am I missing that I can use in finding these $X$ and $Y$?
I have been able to do it when $A=C = \text{Id}_{\mathbb{R}^n}$, where I can choose $Y_{ij} := M_{ij}$ for $i>j$ (and thus $Y_{ij} = -Y_{ji}$ for $i<j$) and then $C=M-Y$ is upper diagonal, but nothing else comes to mind in the general case.
I instead proved that the differential map is injective. Assume $$f_{*(A,C)}(X,Y)=XC+ AY = 0$$ Then, realizing $YC^t + CY^t=0$ from $Y\in T_C O(n)$ and then one expresses $$ Y = A^{-1}XC$$ and then plug it into the condition for $Y$: $$ A^{-1}X C C^t + C (A^{-1}XC)^t = 0$$ yielding with $CC^t=\text{Id}$ $$ A^{-1} X + (A^{-1} X)^t =0$$ meaning $A^{-1}X$ is skey-symmetric. Since both $A$ and $X$ are upper triangular matrices this is only possible if $A^{-1}X = 0$. Then $X=0$ and thus $Y=0$.