Supposing $A$ is an $m \times n$ matrix where $m > n$ and $A$ has full column rank. I want to find a $C$ (an $m \times m$ matrix) such that $A^TCA$ is a diagonal matrix and also that the maximum singular value of $C$ is the smallest possible.
EDIT: $C$ also has to satisfy: $C= VDV^T$ where $V$ is an orthogonal matrix and $D$ is diagonal with positive entries on the diagonal.
Thanks