Suppose I have a matrix $M$ that is symmetric positive definite of size n, then I decompose it using Cholesky decomposition and get $L$:
$$ M = LL^T $$
Let $L_i$ be the i-th column of L:
$$ L = [L_1, L_2, L_3, ..., L_n] $$
And I want to do the following update for an arbitrary $a, b, i, j$:
$$ N = M+a(L_iL_i^T+L_jL_j^T)+b(L_iL_j^T+L_jL_i^T) $$
And then I want to use the cholesky decomposition in $N$:
$$ N = QQ^T $$
I noticed that this update only changes the k-ith columns such that $k = i$ and $k = j$. That is, $Q$ is equal to $L$ with the exception of the columns $i$ and $j$. I'm trying to prove it, but wasn't able to. I managed to prove it for the first update:
$$ N = M+a(L_iL_i^T+L_jL_j^T) $$
Because we have:
$$ M = LL^T = \sum _{i=1}^n L_iL_i^T $$
And then we would only need to multiply the columns $L_i$ and $L_j$ by $(1+a)$. But I have no ideia why it works with the second factor.