I am working on a question related to sensitivity of linear system, and the following step is difficult to prove.
my work:
$\tilde{a}_{ij}-a_{ij}=a_{ij}\epsilon_{ij}$
then, $\tilde{A}-A=A_{\epsilon}$ with $A_{\epsilon}=(a_{ij}\epsilon_{ij})_{ij}$
Taking norm: $||\tilde{A}-A||=||A_{\epsilon}||$
it turns out to prove $||A_{\epsilon}||\leq\epsilon_M||A||$ (for any p-norm)
As side note, This make sense: say in norm-2, it is related to max stretch of transformation, as columns vector of $A_\epsilon$ (i.e. where basis lands after transformation) has max length when each $\epsilon_{ij}=\max{\epsilon}=\epsilon_M$
My guess approach is move from $\epsilon_{ij}$ to $|\epsilon_{ij}|$ then to $\epsilon_M$ i.e. $||A_{\epsilon}||\leq||A_{|\epsilon|}||\leq\epsilon_M||A||$ with $A_{|\epsilon|}=(a_{ij}|\epsilon_{ij}|)_{ij}$
But dealing from the basic definition of p-norm got me totally a mess in proving this
any help would be appreciated