Let $Y \in \mathbb{R}^{n \times n}$ be any matrix such that $Y^T D Y = I$ for some positive diagonal matrix $D$ and $I$ the identity matrix. Further it is known that $Y=X(X^TDX)^{-1/2}$ for some matrix $X \in \mathbb{R}^{n \times k}$.
$Y$, $D$ and $k$ are known.
Is there any way to analytically, or at least numerically, solve for $X$?
Necessarily $k=n$ and $Y,X,D$ are invertible. If $U=D^{1/2}Y$, then $U^TU=I$ and $U$ is an orthogonal matrix. One has $D^{-1/2}U(X^TDX)^{1/2}=X$. Moreover $X^TDX=V^TV$ where $V=D^{1/2}X$; we deduce that $(V^TV)^{1/2}=U^TV$ where the unknown is the invertible matrix $V$. Thus $U^TV=S$ is a symmetric matrix; therefore $V=US$ where $S$ is any invertible symmetric matrix. Finally $X=YS$ where $S$ is an arbitrary invertible symmetric matrix.