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 $z.$