In this lecture prof. Kutz gives the Dynamic Mode Decomposition modes:
$\Phi = X' V_r \Sigma_r^{-1} W $
which are the eigenvectors of the linear propagator matrix. This results from splitting the full data matrix $X$ into $X' = X(:,2:\text{end})$ and $X_1 = X(:,1:\text{end}-1)$. $U_r$, $\Sigma_r$ and $V_r$ are matrices resulting from the SVD of the data matrix $X_1$ and rank-$r$ truncation afterwards. $W$ is the approximation to the eigenvectors of the linear propagator matrix.
However, in the textbook by prof. Kutz Data-driven modeling and scientific computation, this formula is given for the DMD modes:
$\Phi = U_r W$
and this is also the formulation that was originally given by Peter Schmid.
I don't see them being equivalent equations, but perhaps I overlook something. To my understanding, they would be equivalent if in the first equation we used the rank-$r$ approximation matrix $X_{1 r} \approx X_1$ instead of $X'$:
$X_{1 r} V_r \Sigma_r^{-1} W = U_r \Sigma_r V_r^T V_r \Sigma_r^{-1} W = U_r W$
since $V_r$ is both orthogonal and orthonormal, and $\Sigma_r$ is a diagonal matrix.
Could you please explain the logic/intuition behind both approaches and if and how these two approaches differ from each other?