A reminder and notations for the singular value decomposition (SVD) and the Eckart-Young theorem:
Let the SVD of a matrix $A \in \mathbb{R}^{n\times m}$ be $UDV^T$, where $U \in \mathbb{R}^{n\times n}$ and $V \in \mathbb{R}^{m\times m}$ are orthogonal matrices, and $D \in \mathbb{R}^{n\times m}$ such that the values on the main diagonal of $D$ are non-negative and ordered in decreasing order ($\sigma_1 \ge \sigma_2\ge \cdots \ge \sigma_m \ge 0$) and the other entrances of $D$ are zeros.
Let $1 \le r \le m$ and denote $A_r = UD_rV^T$, where $D_r$ is $D$ with the values $\sigma_{r+1}...\sigma_{m}$ replaced by zeros.
The Eckart-Young theorem states that:
$$ A_r = \underset{B\ :\ rank(B)=r}{\mbox{argmin}} \| A - B \|_F $$
where $\| \cdot \|_F$ is the Frobenius norm. The above immediately implies that given $A_{r_1}$ and $A_{r_2}$ such that $r_1 < r_2$, it holds that
$$ \| A - A_{r_1} \|_F > \| A - A_{r_2} \|_F$$
Similarly, due to the uniqueness of the SVD, denoting $U'D'V'^T$ as the SVD of $A_{r_2}$, then $A_{r_1}$ can be derived from $A_{r_2}$ by zeroing the $r_2 - r_1$ smallest values on the main diagonal of $D'$, namely: $$A_{r_1} = U'D'_{r_1}V'^T$$ ($D'_{r_1}$ is defined in the same way as $D_r$ above just with $r_1$.)
Are there any other connections between $A_{r_1}$ and $A_{r_2}$?