C/Multiplication/Real/Angle-preserving/Example

Let

${\displaystyle \varphi \colon \mathbb {C} \longrightarrow \mathbb {C} }$

be a ${\displaystyle {}\mathbb {C} }$-linear mapping that is given by the multiplication with the complex number

${\displaystyle {}w=a+b{\mathrm {i} }\neq 0\,.}$

With respect to the real basis ${\displaystyle {}1,{\mathrm {i} }}$ of ${\displaystyle {}\mathbb {C} =\mathbb {R} ^{2}}$, this mapping is described by the real ${\displaystyle {}2\times 2}$-matrix

${\displaystyle {\begin{pmatrix}a&-b\\b&a\end{pmatrix}}.}$

We write this matrix as

${\displaystyle {}{\begin{pmatrix}a&-b\\b&a\end{pmatrix}}={\begin{pmatrix}{\sqrt {a^{2}+b^{2}}}&0\\0&{\sqrt {a^{2}+b^{2}}}\end{pmatrix}}{\begin{pmatrix}{\frac {a}{\sqrt {a^{2}+b^{2}}}}&-{\frac {b}{\sqrt {a^{2}+b^{2}}}}\\{\frac {b}{\sqrt {a^{2}+b^{2}}}}&{\frac {a}{\sqrt {a^{2}+b^{2}}}}\end{pmatrix}}\,.}$

Therefore, we have a composition of an isometry (a plane rotation) and of a homothety with the scalar factor

${\displaystyle {}\vert {w}\vert ={\sqrt {a^{2}+b^{2}}}\,;}$

in particular, this is an angle-preserving mapping.