Let $A \in \mathbb{C}^{n \times n}$ be invertible and $P$ be a permutation matrix such that
$$PA = \begin{pmatrix}A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix} $$
Where $A_{11} \in \mathbb{C}^{n \times n}$, such that $1 \leq k < n$, is also invertible.
I want to show that if all the elements of $A_{21}A_{11}^{-1}$ have an absolute value that is less than or equal to one, then
$K_{\infty}(A_{22}-A_{21}A_{11}^{-1}A_{12}) \leq n K_{\infty}(A)$
Where $K_{\infty}(X) = ||X||_{\infty}||X^{-1}||_{\infty} $ is the condition number of $X$ with respect to the infinity norm.
$\textbf{Proof Attempt:}$
I believe that I need to write $PA$ in terms of its $LU$ factorization, such that
$$\begin{pmatrix}A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix} = \begin{pmatrix}L_{11} & 0 \\ L_{21} & L_{22} \end{pmatrix}\begin{pmatrix}U_{11} & U_{12} \\ 0 & U_{22} \end{pmatrix}$$
This gives $$A_{11}=L_{11}U_{11} \\ A_{12}=L_{11}U_{12} \\ A_{21}=L_{21}U_{11} \\ A_{22}=L_{21}U_{12} + L_{22}U_{22} $$
This implies $$K_{\infty}(A_{22}-A_{21}A_{11}^{-1}A_{12}) = K_{\infty}(L_{22}U_{12})$$
By the hypothesis, we can also say $||A_{21}A_{11}^{-1}||_{\infty} = ||L_{21}L_{11}^{-1}||_{\infty} < n$, by our assumption and because the infinity norm is equivalent to the maximum absolute row sum.
This is the point where I get stuck, since I don't see how I can get the inequality. I don't see how to to apply the property of the matrix A_{21}A_{11}^{-1} to the problem.
Check out the block inversion formulas and note that if $X$ is a submatrix of $Y$ then $\|X\|_\infty\leq\|Y\|_\infty$. Let $S:=A_{22}-A_{21}A_{11}^{-1}A_{12}$ and let me drop the permutation matrix $P$ here (w.l.o.g.).
Since $S^{-1}$ is the $(2,2)$ block of $A^{-1}$, we have $\|S^{-1}\|_\infty\leq\|A^{-1}\|_\infty$. We also have $$ S=[-A_{21}A_{11}^{-1},I]\begin{bmatrix}A_{12}\\A_{22}\end{bmatrix} $$ and hence $$ \|S\|_\infty = \|[-A_{21}A_{11}^{-1},I]\|_\infty\left\|\begin{bmatrix}A_{12}\\A_{22}\end{bmatrix}\right\|_\infty \leq \|[-A_{21}A_{11}^{-1},I]\|_\infty\|A\|_\infty. $$ The assumption on magnitude of the entries of $A_{21}A_{11}^{-1}$ implies that $\|[-A_{21}A_{11}^{-1},I]\|_\infty\leq n$ so $\|S\|_\infty\leq n\|A\|_\infty$.