Induced matrix norm

Given a norm $∥ \cdot ∥_{*}$ of $R^{n}$ we define a corresponding induced matrix norm according to the formula

∥ A ∥_{*} = m a x_{x \in B_{*}} ∥ A x ∥_{*},

being $B_{*}$ the unit ball of this norm:

B_{*} = {x \in R^{n} : ∥ x ∥_{*} = 1} .