Relation between the weighted matrix norm and the weights

Question

For a nonsingular matrix $W \in \mathbb{C}^{m\times{}m}$, the weighted vector norm is defined as $||\overrightarrow{x}||_W = ||W\overrightarrow{x}||$. Let $||A||$ denote the induced matrix norm by the original vector norm $||\overrightarrow{x}||$, and $||A||_W$ denote the induced matrix norm by the weighted vector norm $||\overrightarrow{x}||_W$. Prove that if $A \in \mathbb{C}^{m\times{}m}$ then $||A||_W = ||WAW^{-1}||$.

Answer 1

By definition $$ \|A\|_W=\sup_{x\ne 0}\frac{\|WAx\|}{\|Wx\|}=\sup_{x\ne 0}\frac{\|WAW^{-1}(Wx)\|}{\|Wx\|}. $$ But as $W$ is non-singular $$ \big\{Wx:x\in\mathbb R^n\smallsetminus\{0\}\big\}=\big\{y:y\in\mathbb R^n\smallsetminus\{0\}\big\}, $$ and hence $$ \sup_{x\ne 0}\frac{\|WAW^{-1}(Wx)\|}{\|Wx\|}=\sup_{y\ne 0}\frac{\|WAW^{-1}y\|}{\|y\|}=\|WAW^{-1}\|. $$

Answer 2

Sketch of Proof: \begin{align} \|A\|_W&=\sup_{\|x\|_W\leq1}\|Ax\|_W\\ &=\sup_{\|Wx\|\leq1} \|WAx\|\\ &=\sup_{\|y\|\leq1} \|WAW^{-1}y\|\\ \\&=\|WAW^{-1}\| \end{align}