Assuming we have a vector with dimension 5x1 of unit length and we change its dimensions by using with m x 5 sized matrices. Example:
$$ v= \left[ \begin{array}{ccc} v_1\\ v_2\\ v_3\\ v_4\\ v_5\\ \end{array} \right],\qquad A = \left[ \begin{array}{ccc} 1&2&3&5&3\\ 2&5&6&4&11\\ 1&2&7&2&3\\ 4&5&6&2&13\\ 6&12&6&2&3\\ \end{array} \right] $$
What I'm trying to find is while assuming $ \left\lVert v\right\rVert = 1$ , what will be $min \left\lVert Av\right\rVert$ and $max \left\lVert Av\right\rVert$?
You may use SVD to attain the answer. So we have $$\min_{||v||=1}||Av||{=\min_{||v||=1}||UDVv||\\=\min_{||v||=1}||DVv||\\=\min_{||w||=1}||Dw||\\=\min_{w_1^2+\cdots+w_n^2=1}\sqrt{|\lambda_1w_1|^2+\cdots+|\lambda_nw_n|^2}}$$where $\lambda_i$s are the eigenvalues of $A$. So we have$$\min_{||v||=1}||Av||=\min_{i}|\lambda_i|$$similarly $$\max_{||v||=1}||Av||=\max_{i}|\lambda_i|$$