The question is below.
Given $C^T M C = D$ for any D a regular matrix in $R^{m, m}$, preferably a diagonal matrix, and M a positive definite matrix in $R^{n, n}$, $ m \neq n$, is it possible to find C? It emerges from theoretical mechanics.
I thanks in advance.
Let $C\in M_{n,m}$; note that $C^TMC$ is sym. $\geq 0$. Since $D$ is invertible, necessarily $D>0$, $n>m$ and $rank(C)=m$.
$M=P^T\Delta P$ where $\Delta$ is $>0$ diagonal and $P\in O(n)$.
Then $C^TP^T\Delta PC=D=(\Delta^{1/2}PC)^T(\Delta^{1/2}PC)=U^TU$ where $U\in M_{n,m}$ has rank $m$.
A particular solution is $U=\begin{pmatrix}V\\0_{n-m,m}\end{pmatrix}D^{1/2}$ where $V\in O(m)$. The general solution is $U=RD^{1/2}$ where $R\in M_{n,m}$ satisfies $R^TR=I_m$.
Finally $C=P^{-1}\Delta^{-1/2}RD^{1/2}$ where $R$ is an arbitrary pseudo-orthogonal matrix.