I have read that the singular values of any matrix $A$ are non-negative (e.g. wikipedia). Is there a reason why?

The first possible step to get the SVD of a matrix $A$ is to compute $A^{T}A$. Then the singular values are the square root of the eigenvalues of $A^{T}A$. The matrix $A^{T}A$ is a symmetric matrix for sure. The eigenvalues of symmetric matrices are always real. But why are the eigenvalues (or the singular values) in this case always non-negative as well?