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Mathematics-Online course: Linear Algebra - Matrices - Matrix Operations

Maximum Absolute Row Sum Norm

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The row sum norm for matrices

$\displaystyle \Vert A\Vert _{infty} = \max\limits_{i} \sum\limits_j \vert a_{ij}\vert $

is the matrix norm associated with the maximum norm for vectors $ \Vert v\Vert _{infty} = \max\limits_i \vert v_i\vert$.
(Authors: Burkhardt/Höllig/Kreitz)
For $ \Vert x\Vert _{infty}=\max\limits_j\vert x_j\vert =1$ we have

$\displaystyle \Vert Ax\Vert _{infty}=\max\limits_i \left\vert\sum\limits_j a_{ij}x_j\right\vert \leq \max\limits_i \sum\limits_j \vert a_{ij}\vert \,, $

and we obtain

$\displaystyle \Vert A\Vert _{infty} = \max\limits_{\Vert x\Vert=1} \Vert Ax\Vert _{infty} \leq \max\limits_{i} \sum\limits_j \vert a_{ij}\vert $

Assuming the maximum is attained for $ i=k$, equality follows by choosing

$\displaystyle x_j = \operatorname{sign} a_{kj}\,, $

since doing so we have

$\displaystyle \left\vert\sum\limits_j a_{kj}x_j\right\vert = \sum\limits_j \left\Vert a_{kj}\right\vert\,. $

(Authors: Burkhardt/Höllig/Kreitz)
  automatically generated 4/21/2005