Consider $\mathbb{R^2}$ with the norm $||x||_p=\left(\sum\limits_{i=1}^n\, |x_i|^p\right)^{\frac{1}{p}}$. Let T be a matrix operator:$T = \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}$ mapping ($\mathbb{R^2},||x||_p)$ into ($\mathbb{R^2},||x||_p)$.
a. Compute $||T||$ when $b=c$ and $p=2$.
b. Compute $||T||$ in general.
My Attempt:
Here is my attempt:
a. I now understand the definition of operator norm and induced matrix norm. I have figured out the answer for part a: The answer should be the maximum eigenvalue of :$ \begin{pmatrix} a & b \\ b & d \\ \end{pmatrix}$ *$ \begin{pmatrix} a & b \\ b & d \\ \end{pmatrix}$ .
which can be easily computed.
b. I'm not sure about part b., however, so any help would be much much appreciated for this part (part b). I understand that there is a simple way of computing the operator norm or induced matrix norm when $p=1$, $p=2$ or when $p=\infty$. However, it is unclear to me what it means to compute $||T||$ in general. I'm not sure if this turns into a lagrange multipliers question using the definition of operator norm, which is:
Thank you very much. The bounty is placed/will be accepted for help on part b.