lu decomposition of submatrix

Question

We have a matrix $A$ and $A'$ is a submatrix of $A$. first we make LU decomposition of $A$, $$A = L_a \cdot U_a.$$ Now, I want to make LU decomposition of $A'$, $$A' = L_a' \cdot U_a'.$$ is there any relation between $L_a, U_a, L_a',U_a'$? Maybe some similar operations between the two LU decompositions?

Answer 1

For an arbitrary submatrix of $A$, the $LU$ decomposition may not exist. This is because $LU$ decompositions only exist for invertible matrices, and arbitrary submatrices of a matrix need not be invertible. Therefore, a general formula will not be possible.