I understand any rectangular mxn matrix A is a factorization of the form:
$$ A = U\Sigma V^{\top}$$
where U and V are orthogonal matrices and $\Sigma$ is diagonal.
Say, I have found U and $\Sigma$ and looking for V.
$$
A = U\Sigma V^{\top} \\
AV = U\Sigma \\
Av_i = \sigma_i u_i \\
A^{\top}Av_i = A^{\top}\sigma_i u_i \\
v_i = A^{\top}\sigma_i u_i \\
v_i = \sigma_iA^{\top}u_i \\
$$
Also from my notes I have,
$$
A = U\Sigma V^{\top} \\
A^{\top}A = A^{\top}U\Sigma V^{\top} \\
I = A^{\top}U\Sigma V^{\top} \\
A^{\top}U = V\Sigma^{\top} \\
(A^{\top}U)_i = v_i\sigma_i \\
v_i = \frac{(A^{\top}U)_i}{\sigma_i}
$$
What am I doing wrong to get these conflicting equations?