# State matrix norm (respect to Matrix) with matrix norm (respect to infinity)

$$\begin{array}{l}{\text { We have following matrix }} \\ {\qquad T=\left(\begin{array}{cc}{-3} & {1} \\ {1} & {2}\end{array}\right)} \\ {\text { a) Show that, with }\|x\|_{T}=\|T x\|_{\infty} \text { a vector norm is defined for } \mathbb{R}^{2} \text { }} \\ {\text { (Hint: Properties of a norm.) }}\end{array}$$ $$\begin{array}{l}{\text { b) Sketch the unit circle } B_{T}=\left\{x \in \mathbb{R}^{2}\|x\|_{T} \leq 1\right\}} \\ {\text { c) State the }\|\cdot\| T \text { assigned matrix norm }} \\ {\quad\|A\|_{T}:=\max _{x \neq 0} \frac{\|A x\|_{T}}{\|x\|_{T}}} \\ {\text { with the matrix norm}\|\cdot\|_{\infty} \text { }}\end{array}$$

Trying to do c) my hint is to reform it until there is only a maxx norm but how do i do that?

Hint. Try to rewrite $$\max_{x\ne0}\frac{\|Ax\|_T}{\|x\|_T}$$ in the form of $$\max_{y\ne0}\frac{\|By\|_\infty}{\|y\|_\infty}$$ where $$B$$ is some matrix that depends on $$T$$ and $$A$$, and $$y$$ is some vector depending on $$T$$ and $$x$$. It follows that $$\|A\|_T=\|B\|_\infty$$.
! For every nonzero vector $$x$$, let $$y=Tx$$. Then $$\frac{\|Ax\|_T}{\|x\|_T} =\frac{\|TAx\|_\infty}{\|Tx\|_\infty} =\frac{\|TAT^{-1}y\|_\infty}{\|y\|_\infty}.\tag{1}$$ Since $$T$$ is invertible, there is a one-to-one correspondence between $$x$$ and $$y$$. Therefore, the maximum of the LHS of $$(1)$$ over all nonzero vectors $$x$$ is equal to the maximum of the RHS over all nonzero vectors $$y$$. It follows that $$\|A\|_T=\max_{x\ne0}\frac{\|Ax\|_T}{\|x\|_T} =\max_{y\ne0}\frac{\|TAT^{-1}y\|_\infty}{\|y\|_\infty} =\|TAT^{-1}\|_\infty.$$
• $$\|\mathbf{x}\|_{T} = \|T x\|_{\infty}$$ but what's with the $$\|A x\|_{T}$$ – Rack Cloud Nov 5 '19 at 1:07
• Yeah i'm not that good in math trying to get better $$\mid \mid Ax \mid \mid_{T} = \mid \mid TAx \mid \mid_{inf}$$ is there any good site/video to get better with that norm formulations and equations? Are these equations always valid? – Rack Cloud Nov 5 '19 at 9:54