$\mathbf{X} \in \mathbb{R}^{M\times N}$, $M$ is the number of data, $N$ is the dimension of data. Then one can have SVD as $\mathbf{X= U\Sigma V^\top} $.
However, now I do the SVD on the tranlsated and scaled $\mathbf{X}$, $$ Y = (\mathbf{X} - \begin{bmatrix} 1 \\ \vdots \\ 1 \end{bmatrix} \mathbf{\mu} ) \mathbf{\Lambda}^{-1} $$ where $\mu$ is the mean of the data. $\Lambda$ is the standard deviation of the data for each dimension.
My question is, what is the relationship between the following SVD On $Y$ and $X$? $$ Y = U_Y \Sigma_Y V_Y^\top $$
For my own interests, is there any closed form relationship between $V_Y$ and $\mathbf{V}$?