Consider the following system of equations in $\mathbb{R}^n$:
$y = Cx +b$,
where $C$ is an $n\times n$ matrix with generic element $(c_{j,k})$. This matrix C satisfies $\sum_{k=1}^{n} |c_{j,k}| <1$, for any $j$ and $b \in \mathbb{R}^{n}$ is a given constant vector.
I'm asked to show that there's a solution for this system in the form
$x = Cx +b$.
This seems like an exercise about contractions and finding a fixed point. Here's my work:
Let $T(x) = Cx + b$. If we establish that $T$ is a contraction, then we have our result, because it's defined over a complete domain.
So: \begin{gather} \| T(x) - T(y)\| =\|C(x-y)\| = \sqrt{\sum_{j}(\sum_{k}|c_{j,k}|\left(x_{k} - y_{k}\right))^2} \leq \|x-y\|\sqrt{\sum_{j}\sum_{k}c_{j,k}^2} \end{gather}
Sure, I can invert limits and use that the sum in $j$, for each $k$ will be less than $1$. My problem is that all upper bounds I can find for this depend on $n$, or $\sqrt{n}$ to be precise and, hence, I can't affirm that I have a contraction. What am I missing here?