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?
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Use the formula for generating all Pythagorean triples and choose the matrix accordingly. Everything is explained here with $3\times 3$ matrices, section "Pythagorean triples by use of matrices and linear transformations". If you prefer a link to a duplicate on MSE, then look here, and the references given there. – Dietrich Burde Oct 06 '16 at 09:08
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thanks for your comment, but that does not answer the question how one can come up with this matrix – Martin Oct 13 '16 at 07:44
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It does. Just rewrite the triple-generating formula as matrix multiplication, see the wikipedia section "Pythagorean triples by use of matrices and linear transformations". How to find the formula, see here, i.e., here. – Dietrich Burde Oct 13 '16 at 07:50