Let $A$ and $B$ be symmetric positive definite (SPD) matrices and $C=A+B$. I know the SVD or Cholesky decomposition of A and B, $A=U_A\Sigma_AU_A^T=L_AL_A^T$ and $B=U_B\Sigma_BU_B^T=L_BL_B^T$.
Can I use the factors from $A$ and $B$ to directly compute the SVD or Cholesky decomposition of $C$? Or do I have to compute $C=A+B$ and then do either SVD or Cholesky on $C$ directly?
This is relatable to QR decomposition or SVD of the sum of matrices, but in my case $A$ and $B$ are SPD.