The contraction transformations of \(\RNrSpc{n}\) are given by
\[C\from \RNrSpc{n} \longrightarrow \RNrSpc{n},\quad C(\Vect{x}) \DefEq s\cdot\Vect{x}\]where \( 0 < s < 1 \) is a fixed number. – Why is it called a ‘contraction transformation’? – \(C\) is called a contraction transformation because it shrinks or contracts each distance and, therefore, each object in \(\RNrSpc{n}\) by the factor \(s\).
To find the matrix representing \(C\), we note that \(C\) transforms the basic vector \(\StdBssVec{j}\) into
\[C(\StdBssVec{j}) = s\cdot \StdBssVec{j} = (0,\dots ,0,s,0,\dots ,0)\]Therefore \(C\) is represented by the matrix of size \((n,n)\)
\[ s\cdot \IdMtrx{n} = \left[ \begin{array}{cccc} s & 0 & \cdots & 0 \\ 0 & s & & 0 \\ \vdots & & \ddots & \vdots \\ 0 & 0 & \cdots & s \end{array} \right] \]For example, the dilation of \(\RNrSpc{2}\) in the picture above is
\[ D\from \RNrSpc{2} \longrightarrow \RNrSpc{2},\qquad D(x,y)=\tfrac{3}{2}\cdot (x,y) = \left[ \begin{array}{cc} \tfrac{3}{2} & 0 \\ 0 & \tfrac{3}{2} \end{array} \right] \left[ \begin{array}{c} x \\ y \end{array} \right] \]