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Plane similarities are complex or conjugate-complex multiplications; the orientation-preserving ones are exactly the nonzero complex multiplications

Statement

Let $L : R^{2} \to R^{2}$ be real-linear. It is a similarity if and only if exactly one of the following forms holds for some $ξ \in C ∖ {0}$ :

$L h = ξ h, L h = ξ \overset{ˉ}{h} .$

The first form is orientation-preserving and the second orientation-reversing. Thus the orientation-preserving similarities are exactly the nonzero complex multiplications.

Facts & Assumptions

Given: A real-linear map $L : R^{2} \to R^{2}$ .

[F1]

A similarity has a ratio $λ > 0$ and satisfies $⟨ L h, L k ⟩ = λ^{2} ⟨ h, k ⟩$ ; its orientation is the sign of $ω (L e_{1}, L e_{2})$ (Orientation-preserving conformality for a real-differentiable complex map at a point).

[F2]

The Euclidean inner product is $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ and is positive definite (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[L1]

Under $C ≅ R^{2}$ , multiplication satisfies $(a + b i) (x + i y) = (a x - b y) + i (b x + a y)$ ( $C$ is the real coordinate plane, with coordinate arithmetic).

Proof

technique · direct

1.1

Suppose $L$ is a similarity of ratio $λ$ , and write its columns as $p = L e_{1} = (a, b)$ and $q = L e_{2} = (c, d)$ . By [F1]–[F2], $p ⊥ q$ and $∣ p ∣ = ∣ q ∣ = λ > 0$ .

givenF1F2

1.2

Conversely, for $ξ = a + b i \neq 0$ , direct expansion using [F2] shows that both $h \mapsto ξ h$ and $h \mapsto ξ \overset{ˉ}{h}$ multiply every inner product by $∣ ξ ∣^{2}$ . Their signed area factors are respectively $∣ ξ ∣^{2}$ and $- ∣ ξ ∣^{2}$ , so both are similarities with the asserted orientations.

F1F2L1algebra

2.1

In the plane, a vector orthogonal to the nonzero $p = (a, b)$ and of the same length is either $(- b, a)$ or $(b, - a)$ . Hence $q$ is one of these two vectors.

step 1.1algebra

3.1

If $q = (- b, a)$ , [L1] gives $L h = (a + b i) h$ and $ω (p, q) = a^{2} + b^{2} > 0$ . If $q = (b, - a)$ , [L1] gives $L h = (a + b i) \overset{ˉ}{h}$ and $ω (p, q) = - (a^{2} + b^{2}) < 0$ .

step 2.1L1algebra

4.1

The signed area factor cannot be both positive and negative, so the two forms are mutually exclusive and the classification is complete.

step 3.1step 1.2algebra∎

Depends on

Used by

A real-differentiable complex map is orientation-preserving conformal at a point exactly when it is complex differentiable there with nonzero derivative Theorem

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. Lebl, Guide to Cultivating Complex Analysis, Proposition 2.2.9 and Exercises 2.2.22–23 (standard reference, not scraped)