Is only one SVD computation enough to perform PCA on a matrix and its transpose?

SVD is used in PCA in order to get the mapping to lower dimensions. Is it enough to perform only one SVD in order to get the PCA for the original matrix AND its transpose, considering that the SVD of a matrix gives also the SVD of the transpose of that same matrix?

SVD decomposition of $Y$ and $Y^t$:

$$Y=USigma V^t$$
$$Y^t=VSigma^tU^t$$

Solution of PCA for matrix Y:
$$vec{x}=W^tvec{y}=U_k^tvec{y}$$