Pythagorean Generator Matrix

Number Theory Level 2

\[\begin{pmatrix}x & y & y \\ y & x & y \\ y & y & z \end{pmatrix}\begin{pmatrix}a \\ b \\ c \end{pmatrix} = \begin{pmatrix}a' \\ b' \\ c' \end{pmatrix}\] If the above equation holds true, then there exist coprime positive integers \(x, y, z\) such that any Pythagorean triple \((a, b, c)\) produces another Pythagorean triple \(\big(a', b', c'\big).\)

Find the sum of such \(x, y,\) and \(z.\)