In Heisenberg's [*The Physical Principles of the Quantum Theory*](https://archive.org/stream/in.ernet.dli.2015.166273/2015.166273.The-Physical-Principles-Of-The-Quantum-Theory#page/n147) he has a section of Perturbation Theory, where he develops Perturbation theory on the Matrix Theory he's developed in the earlier sections of the book. It goes accordingly:

There is the Hamiltonian $H$ and a matrix $S$, which transforms the Hamiltonian into a coordinate system in which the Hamiltonian is diagonal, in which it is represented by $W$. As such:
$$S^{-1}HS=W$$
or
$$HS=SW$$
He then proceeds to develop the perturbed Hamiltonian, Energy and Transformation Matrices, with $\lambda$ as a dummy variable
$$H=H\_0+\lambda H\_1+\lambda^2H\_2+...$$
$$W=W\_0+\lambda W\_1+\lambda^2W\_2+...$$
$$S=S\_0+\lambda S\_1+\lambda^2S\_2+...$$
He then says that by replacing them in $HS=SW$ and setting equal the terms with equal $\lambda$ coefficients he obtains
$$\begin{matrix}
H\_0S\_0=S\_0W\_0\\
H\_0S\_1-S\_1H\_0=S\_0W\_1\\
H\_0S\_2-S\_2H\_0+H\_2S\_2-S\_1W\_1=S\_0W\_2\\
\vdots
\end{matrix}
$$
But these are not the results I obtain. The results I obtain are
$$\begin{matrix}
H\_0S\_0=S\_0W\_0\\
H\_0S\_1+H\_1S\_0=S\_0W\_1+S\_1W\_0\\
H\_0S\_2+H\_2S\_0+H\_1S\_1=S\_0W\_2+S\_2W\_0+S1W\_1\\
\vdots
\end{matrix}$$
Please help.