Applying the Banach's Contraction Principle

Question

I have a problem:

For a system of linear equations: $$x_i=\sum_{j=1}^{n}a_{ij}x_j+b_i,\ \ i=1,2, \ldots , n \tag 1$$ Prove that, if
$$\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}^2 \le q<1$$ then $(1)$ has a unique solution.

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My teacher said that We need to use the Banach's Contraction Principle, but I have trouble when I do it...

Any help will be appreciated! Thanks!

Answer 1

Define $\pmb{A}$ to be the matrix with entries $a_{i,j}$ for $i,j = 1, \ldots, n$, $\pmb{x} = (x_1, \ldots, x_n)$ and $\pmb{b} = (b_1, \ldots, b_n)$. Then your system is equivalent to $$\pmb{x} = \pmb{A}\pmb{x} + \pmb{b}$$ Define $f: \mathbb{R}^n \to \mathbb{R}^n$ by $f(\pmb{x}) = \pmb{A}\pmb{x} + \pmb{b}$. Equip $\mathbb{R}^n$ with the Euclidean metric.

What does it mean for $f$ to be a contraction in this metric space, and what does Banach's theorem say in that case?

Answer 2

The inequality you have derived holds in a given norm.
Now you do not have to show explicitly, that $|| A|| \leq \sum_{i,j} a_{ij}^2$ holds. The double sum which you are given is a specific norm(Hilbert-Schmid or Frobenius) and what you essentially have yet to show to be done with the task is that it is consistent with the euclidean norm for vectors.