If you're interested in learning more about SVD, I suggest referencing the book "Numerical linear algebra" by Trefethen and Bau (should be available online through the CMU library). The SVD wiki page is good too.
To compute the SVD, we can also reduce the SVD computation problem to an eigenvector problem. That is, consider the matrix $A^T A = (U \Sigma V^T)^T (U \Sigma V^T) = V \Sigma^2 V^T$. The eigenvectors of $A^T A$ therefore correspond to $V$, and the eigenvalues of $A^T A$ are the square of the singular values in $\Sigma$. So if you compute the eigenvalues/eigenvectors of $A^T A$, it is fairly straightforward to recover the singular values and vectors of the original matrix $A$.
If you're interested in learning more about SVD, I suggest referencing the book "Numerical linear algebra" by Trefethen and Bau (should be available online through the CMU library). The SVD wiki page is good too.
To compute the SVD, we can also reduce the SVD computation problem to an eigenvector problem. That is, consider the matrix $A^T A = (U \Sigma V^T)^T (U \Sigma V^T) = V \Sigma^2 V^T$. The eigenvectors of $A^T A$ therefore correspond to $V$, and the eigenvalues of $A^T A$ are the square of the singular values in $\Sigma$. So if you compute the eigenvalues/eigenvectors of $A^T A$, it is fairly straightforward to recover the singular values and vectors of the original matrix $A$.