I've thought about this problem for days but could not find a good answer.
Given Cholesky decomposition of a symmetric positive semidefinite matrix $A = LL^T$. Now, suppose that we delete the $i$-th row and the $i$-th column of $A$ to obtain $A'$ ($A'$ is also a symmetric positive semidefinite matrix), and the Cholesky decomposition of this new matrix is $A' = L'(L')^T$.
Is there any efficient way to obtain $L'$ from $L$?
Note: In case we delete the last row and the last column of $A$, the problem becomes simple, we just delete the last row and last column of $L$ to obtain $L'$. Other cases, for me, are not trivial.
Thank you so much.