How does MATLAB, for instance, calculate the SVD of a given matrix? I assume the answer probably involves computing the eigenvectors and eigenvalues of A*A'. If that is the case, I would also like to know how does it compute those?
How is the SVD of a matrix computed in practice
linear algebra;matrices
This is typically done using the Golub-Reinsch algorithm, and no, it doesn't involve computing eigenvalues and eigenvectors of $AA^{T}$.  See G. H. Golub and C. Reinsch.  Singular Value Decomposition and Least Squares Solutions.  Numerische Mathematik 14:403-420, 1970.This material is discussed in many textbooks on numerical linear algebra.  