Compute $\|T\|_1$, $\|T\|_2$ and $\|T\|$

Question

I came across the following problems but I had no idea how to solve this. Any ideas would be appreciated! Let $T \in L(C^n)$ be given by $Tx = Ax$ where $A$ is the $n\times n$ matrix with all entries equal to $1$. How do we compute $\|T\|_1$, $\|T\|_2$ and $\|T\|$?

Answer 1

Did you lookup the definitions of these norms? I'll assume that you meant $\|T\|_{\infty}$ instead of just $\|T\|$.

$$ \|A\|_1 = \max_{j=1,\cdots, n} \sum_{i=1}^n|A_{ij}|= \max\{n, n, n, \cdots, n\}=n $$

$$ \|A\|_{\infty} = \max_{i=1,\cdots, n} \sum_{j=1}^n|A_{ij}|= \max\{n, n, n, \cdots, n\}=n $$

Finally, $\|A\|_2$ is the spectral norm, which in this case will be the maximum eigenvalue (in absolute value), also $n$. This is one of those rare examples in which all these norms coincide.