Reading "Multiple View Geometry in Computer Vision", I am struggling to understand a result made to approximate a matrix $F$ to the closest singular matrix $F'$ (which represents the $3\times3$ Fundamental Matrix).
This Fundamental matrix $F'$ has to be singular (more concretely of rank $2$). However, due to noise the initial guess $F$ may be of rank $3$. Given this, the authors state that:
Let $F = UDV^T$ be the SVD of $F$, where $D$ is a diagonal matrix $D = \text{diag} (r, s, t)$ satisfying $r ≥ s ≥ t$. Then $F'= U\text{diag(r, s, 0)}V^T$ minimizes the Frobenius norm of $F − F$
But I can't understand why just by setting the smaller singular value of $F$ to zero we reach the matrix $F'$ (subject to $\det(F')=0$) that minimizes $\Vert F - F' \Vert^2$. Besides, why the same singular vectors are needed?
Thanks in advance!