The norm for our matrix is defined as:
$$ \|A\| = \max_{x\neq 0} \frac{\|A\pmb x\|_q}{\|\pmb x\|_p} $$
I'm asked to "characterize" the matrix norm for various situations.
Here was my approach (you'll see in Q3 that I noticed a flaw in my reasoning that probably invalidates my answers for Q1 and Q2)
Q1 $q=p=\infty$
$$ \begin{align} \|A\| &= \max_{x\neq 0} \frac{\|A\pmb x\|_1}{\|\pmb x\|_1}\\[10pt] &= \frac{\|(\pmb u \Sigma \pmb v^T)(a \pmb{\bar x})\|_1}{\|a \pmb{\bar x}\|_1}\\[10pt] &= \frac{|a|}{|a|} \frac{\|\pmb u \Sigma \pmb (v^T\pmb{\bar x})\|_1}{\|\pmb{\bar x}\|_1}\\ &= \frac{\|\pmb u \Sigma(1))\|_1}{\| \pmb{\bar x}\|_1}\\ &= \frac{\|\begin{pmatrix} \lambda_1 & 0 & \cdots & 0\end{pmatrix}\|_1}{\sum_{i=1}^n |{\bar x}_i |} \\ &= \frac{|\lambda_1|}{\sum_{i=1}^n |{\bar x}_i |} \end{align} $$
Q2 $q=p=1$
Follow the same steps before except at the end we have: $$ \frac{|\lambda_1|}{\max(\bar x_i)} $$
Q3 $q=\infty$ and $p=1$
When thinking about question 3 I assumed that orienting vector $\pmb x$ didn't grow the norm in the denominator any more than moving the direction of stretch to the largest singular value...
Which then leaves me stumped on all $3$.
Any suggestions for how to approach the problem? I can probably just give a lower bound if there isn't a better way to go about it.