In theorem 15.14 of Transformation Geometry — An Introduction to Symmetry, to prove that an affine transformation is a product of strains, George E. Martin takes a shear mapping
$$\begin{cases}x'=x+y \\ y'=y\end{cases},$$
"casts a spell", and the mapping decomposes into a composition of a similarity, a strain, and a similarity, arising from the decomposition of its matrix $$\begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}= \begin{pmatrix}\frac{5-\sqrt{5}}{20} & \frac{5-3\sqrt{5}}{20} \\ -\frac{5-3\sqrt{5}}{20} & \frac{5-\sqrt{5}}{20}\end{pmatrix} \begin{pmatrix} \frac{3+\sqrt{5}}{2} & 0 \\ 0 & 1\end{pmatrix} \begin{pmatrix} 2 & 1+\sqrt{5} \\ -(1+\sqrt{5}) & 2\end{pmatrix}.$$ Is there a way to arrive to this decomposition (either of the matrix or of the mapping) naturally?