Substitution matrix for the Ammann-Beenker tiling On the wikipedia page for the Ammann-Beenker tiling, it says the following:

I am trying to derive the substitution matrix for and show that its eigenvalues are $(1\pm\sqrt{2})^1$... so far the things I found online all scale as $(1\pm\sqrt{2})^2$.
Where would I start?
 A: The matrix of the 1D silver-mean substitution $R \to RrR, r \mapsto R$ is
$$
\begin{bmatrix}
2 & 1\\
1 & 0
\end{bmatrix}
$$
whose eigenvalues are the silver mean and its PV conjugate $1 \pm \sqrt{2}$.
For the Amman-Beenker substitution, the substitution matrix on the tiles is
$$
\begin{bmatrix}
3 & 2\\
4 & 3
\end{bmatrix}
$$
(where I've chosen to cut the squares into two triangles in order to have a proper Stone inflation - you could also treat these triangles separately if you really wanted, but it doesn't change the calculation as they have the same area and substitute dually)
whose eigenvalues are $(1+\sqrt{2})^2$ and its algebraic conjugate. This makes sense as the Perron-Frobenius eigenvalue of the substitution matrix corresponds to the inflation factor as it acts on area so it inflates a region of the plane with area $A$ to a region with area $A(1+\sqrt{2})^2$. Accordingly, as the substitution only expands, and there is no shearing, the linear expansion factor for lengths is just the square root of the area expansion factor, hence $1+\sqrt{2}$.
