Question on Rank Constraint

I want to make the below constraints.

βˆ‘_π‘Ÿπ‘‘β–’γ€–γ€–π‘†π‘‡γ€—(π‘Ÿπ‘‘,𝑖)βˆ—πΆ(π‘Ÿπ‘‘,𝑙𝑑) γ€—=0
π‘…π‘Žπ‘›π‘˜(C_(π‘Ÿπ‘‘,𝑙𝑑) )= πΆπ‘Žπ‘Ÿπ‘‘(𝑅𝑇) βˆ’ π‘…π‘Žπ‘›π‘˜(〖𝑆𝑇〗_(π‘Ÿπ‘‘,𝑖)) = 12

ST(rt, i) is known matrix,
and C(rt, ld) is unknown matrix (variable) which represents the dependence of row-vector(rt) in ST matrix.

Because rank of the ST matrix is already known, the number of elements in set C is also known (12).

From this aspect, in order to make sure the uniqueness (independence) of columns (ld) in C matrix, I set the rank constraint.
RankΒ© = 12

How do you think of the rank-constraint?

I’m not sure whether or not C matrix is convex.

Non-convex. Why isn't CVX accepting my model? READ THIS FIRST!

Edit: I find your post to be unclear. If C is a symmetric square 12 by 12 matrix, you can impose the constraint lambda_min(C) >= small_number to enforce that C be full rank (12). However, I am guessing this is not your situation.