Definition:Contraction
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Definition
Let $G$ be a vector space over a field $K$.
Let $\beta \in K$.
Let $s_\beta: G \to G$ be a similarity mapping on $G$:
- $\forall \mathbf x \in G: \map {s_\beta} {\mathbf x} = \beta \mathbf x$
Let the scale factor $\beta$ be such that $0 < \beta \le 1$.
Then $s_\beta$ is called a contraction.
Also see
- Results about similarity mappings can be found here.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations: Example $28.3$