MATH661

By definition

{|| 𝑨 ||}_{\infty} = {sup}_{{|| 𝒙 ||}_{\infty} = 1} {|| 𝑨 𝒙 ||}_{\infty},

and with above notation ${|| 𝒙 ||}_{\infty} = | x_{k} | = 1$ . The vector $𝒚 = 𝑨 𝒙$ , $𝑨 = [a_{i j}]$ has components

y_{i} = \sum_{j = 1}^{n} a_{i j} x_{i}

and the inf-norm is

{|| 𝑨 𝒙 ||}_{\infty} = {|| 𝒚 ||}_{\infty} = {max}_{i} | \sum_{j = 1}^{n} a_{i j} x_{j} | .

Consider the vector $𝒙$ with components $x_{j} = sign (a_{i j})$ for which ${|| 𝒙 ||}_{\infty} = 1$ . Then

| \sum_{j = 1}^{n} a_{i j} x_{j} | = \sum_{j = 1}^{n} a_{i j} x_{j} = \sum_{j = 1}^{n} | a_{i j} |

and the maximum is obtained for the row vector of $𝑨$ of greatest 1-norm,

{|| 𝑨 ||}_{\infty} = {max}_{1 ⩽ i ⩽ m} {|| 𝒂_{i}^{*} ||}_{1} .

Consider now the 2-norm definition

{|| 𝑨 ||}_{2} = {sup}_{{|| 𝒙 ||}_{2} = 1} {|| 𝑨 𝒙 ||}_{2}