Given matrices $C\in \mathbb{R}^{n\times k}$ and $X\in \mathbb{R}^{n\times k}$, it is known that $X$ minimizes the following
$\|CC^T-XX^T\|_F^2$
Can it be proved that such solution X minimizes
$\|C-X\|_F^2$
Note that $\|\cdot\|_F$ corresponds to the Frobenius norm.
Edit
The question is as I posted it primarily. Note that the dimensions of $C$ and $X$ need to be the same; otherwise $\|C-X\|^2$ would not be possible. With your $X=CU$, what is obtained is $\|CC^T-CC^T\|$. Note that one could trivially set $X=C$ to annihilate Frobenius norm, but that's not the point. Matrix $X$ is supplied externally, and it is known that it minimizes $\|CC^T-XX^T\|^2$; question is: does it also minimize $\|C-X\|^2$?
In the external algorithm, matrix $X$ is obtained as $X=Lsv(C)SV(C)^{1/2}$, where $Lsv(C)$ denotes left-singular vector of $C$, and $SV(C)^{1/2}$ a diagonal matrix with roots of singular values of C. Note that $||CāX||^2$ does not need to be 0 to be minimal.