I'd like some help with understanding the following statements...I saw it on the internet while searching for a proof, and I'd like to understand why its true:
let $A$ be a diagonally dominant matrix with respect to the columns (meaning $|a_{jj}|$ is larger than the sum of the absolute values of the entries in column $j$).
Then:
$$\sum_{i=2,i\neq j}^{n} |a_{ij}-\frac{a_{1j}a_{i1}}{a_{11}}| \leq \sum_{i=2,i\neq j}^{n}|a_{ij}|+\sum_{i=2,i\neq j}^{n}|\frac{a_{1j}a_{i1}}{a_{11}}| \leq (|a_{jj}-|a_{1j}|)+\frac{|a_{1j}|(|a_{11}|-|a_{j1}|}{|a_{11}}$$
which is equal to
$$|a_{jj}-\frac{a_{j1}a_{1j}}{a_{11}}|$$
Firstly, I don't understand why the first inequality is true. And even if it was, I don't understand how the second one follows (it should follow from $A$ being diagonally dominant), and I don't understand why the last term in the inequality is equal to what I wrote after "which is equal to".
Would really appreciate help understanding this part,