Definition:Similarity Mapping

Definition

Let $G$ be a vector space over a field $K$.

Let $\beta \in K$.

Let $s_\beta: G \to G$ be the mapping on $G$ defined as:

$\forall \mathbf x \in G: \map {s_\beta} {\mathbf x} = \beta \mathbf x$

$s_\beta$ is called a similarity (mapping).

Scale Factor

The coefficient $\beta$ is called the scale factor of $s_\beta$.

Also known as

An older term for a similarity mapping is similitude.

Also see

• Definition:Stretching
• Definition:Contraction

• Similarity Mapping is Linear Operator

• 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 $28.3$