Construction of Pythagorean triples

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If $(x,y,z)$ is a Pythagorean triple, then $\begin{pmatrix} 2 & 1 & 2 \\ 1 & 2 & 2 \\ 2 & 2 & 3 \end{pmatrix} \cdot \begin{pmatrix} x \\ y \\ z \end{pmatrix}$ is again a Pythagorean triple. This is easy to show. But how can one come up with this matrix?