[proofplan] We show bandwidth preservation by induction on the elimination steps. At each step, the multipliers $\ell_{ik}$ vanish for $i > k + r$ (since $A_{ik} = 0$ outside the band), and the rank-one update $\ell_1u_1^\top$ only affects entries within the band, preserving the bandwidth of the remaining submatrix. [/proofplan] [step:Show the first elimination step preserves bandwidth] Partition $A$ as $\begin{pmatrix} a_{11} & u_1^\top \\ \ell_1 a_{11} & A' \end{pmatrix}$. Since $A$ has bandwidth $r$: $A_{i1} = 0$ for $i > r+1$ (so $(\ell_1)_i = 0$ for $i > r$), and $A_{1j} = 0$ for $j > r+1$ (so $(u_1)_j = 0$ for $j > r$). The rank-one update $\ell_1 u_1^\top$ has $(\ell_1 u_1^\top)_{ij} = 0$ whenever $i > r$ or $j > r$. Therefore $A' - \ell_1u_1^\top$ also has bandwidth $r$. [/step] [step:Complete the induction and conclude] By induction, each elimination step preserves bandwidth $r$ of the remaining submatrix. The multipliers $\ell_{ik}$ in column $k$ of $L$ satisfy $\ell_{ik} = 0$ for $i > k + r$ (since $A_{ik}^{(k)} = 0$ outside the band). Similarly, $U_{kj} = 0$ for $j > k + r$. Therefore both $L$ and $U$ have bandwidth $r$. [/step]