Singular Subspaces

A pair of $k$-dimensional subspaces $\cal U$ and $\cal V$ are called (left and right) singular subspaces of $A$ if $Av \in \cal U$ for all $v \in \cal V$ and $A^*u \in \cal V$ for all $u \in \cal U$. We also write this as $A {\cal V} \subset {\cal U}$ and $A^* {\cal U} \subset {\cal V}$.

The simplest example is when $\cal U$ and $\cal V$ are spanned by a single pair of singular vectors $u_i$ and $v_i$ of $A$, respectively. More generally, any pair of singular subspaces can be spanned by a subset of the singular vectors of $A$, although the spanning vectors do not have to be singular vectors themselves.