Assume $A=BB^T$, where $B\in \mathbb{R}^{n\times m}$ and therefore $A\in \mathbb{R}^{n\times n}$. The product $A$ is always symmetric positive definite. I want to find the Cholesky factor $A=LL^T$, where $L\in \mathbb{R}^{n\times n}$ and it is lower triangular.
Is there a way to relate $B$ to $L$ without computing $A$ first?
Edit: Same question, but let now $A=(C\cdot B)B^T$, where $C \cdot B$ is the elementwise multiplication of C and B and $C=[C_1, C_2,...,C_m]$ is a collection of column vectors. Each column vector is a constant, $C_i = c_i I$ where $c_i$ is a scalar and $I\in \mathbb{R}^{n \times 1}$.