Definition
36
[11G] Let \(A∈ℝ^{m× n}\) be a matrix; considering it as a linear application between normed spaces \((ℝ^ n,||_ p)\) and \((ℝ^ m,||_ q)\), let’s define again the induced norm as
\begin{equation} \| A\| _{p,q}{\stackrel{.}{=}}\max _{x∈ℝ^ n~ ,~ |x|_ p≤ 1} |Ax|_ q\label{eq:norme_ matrici} \end{equation}
37
(Note that the maximum is always reached at a point with \(|x|_ p=1\)).