Assume we have a $m \times n$ matrix $A$ with real entries representing an operator $T$ on $n$ dimensional real vector space $V$. Then we select a $n-1$ dimensional subspace of $E$ of $V$ and restrict the $T$ to $E$. Say the matrix representing $T|\_{E}$ is $A\_{E}$. What is the range of singular values of $A\_{E}$ in terms of singular values of $A$? 



range of singular values of sub-matrices
I assume that by singular values, you mean eigenvalues of $A^\*A$ (more precisely, their positive square roots). There are variational characterisations
\begin{align\*}
\lambda\_i&=\min\_{\begin{smallmatrix}U\subset V\\\dim U=i\end{smallmatrix}}\max\_{\begin{smallmatrix}v\in U\\|v|=1\end{smallmatrix}}|Av|\;,\\
&=\max\_{\begin{smallmatrix}U\subset V\\\dim U=n+1-i\end{smallmatrix}}\min\_{\begin{smallmatrix}v\in U\\|v|=1\end{smallmatrix}}|Av|\;.
\end{align\*}
If you restrict to $E\subset V$ and denote the singular values of $A\_E$ by $\mu\_j$,
you get $\lambda\_i\le\mu\_i$ from the first and $\mu\_i\le\lambda\_{i+1}$ from the second equation. So
$$\lambda\_1\le\mu\_1\le\lambda\_2\le\cdots\le\mu\_{n-1}\le\lambda\_n\;.$$


