Row sum norm

Illustration of the line sum norm

The row sum norm is in the mathematics of the maximum norm derived natural matrix norm . The row sum norm of a matrix corresponds to the maximum amount sum of its rows. It is sub-multiplicative and compatible with the maximum norm . The line sum norm is used in particular in linear algebra and numerical mathematics .

definition

The row sum norm of a matrix with as the field of real or complex numbers is the natural matrix norm derived from the maximum norm and is thus defined as ${\displaystyle \|\cdot \|_{\infty }}$ ${\displaystyle A\in {\mathbb {K} }^{m\times n}}$ ${\displaystyle \mathbb {K} }$

${\displaystyle \|A\|_{\infty }:=\max _{x\neq 0}{\frac {\|Ax\|_{\infty }}{\|x\|_{\infty }}}=\max _{\|x\|_{\infty }=1}\|Ax\|_{\infty }}$.

The line sum norm clearly corresponds to the greatest possible stretching factor that results from the application of the matrix to a vector with an entry of one as the maximum amount. The name-giving representation applies to the line sum norm

${\displaystyle \|A\|_{\infty }=\max _{\|x\|_{\infty }=1}\|Ax\|_{\infty }=\max _{\|x\|_{\infty }=1}\max _{i=1,\ldots ,m}\left|\sum _{j=1}^{n}a_{ij}x_{j}\right|=\max _{i=1,\ldots ,m}\max _{\|x\|_{\infty }=1}\left|\sum _{j=1}^{n}a_{ij}x_{j}\right|=\max _{i=1,\ldots ,m}{\sum _{j=1}^{n}|a_{ij}|}}$.

It was used here that the sum within the amount lines for fixed is maximum exactly when is for all . The line total standard is calculated by determining the total amount for each line and then selecting the maximum of these values. The following rule of thumb helps to distinguish it from the related column sum norm : it is vertical and stands for the columns, while the one is horizontal and stands for the rows. ${\displaystyle i}$ ${\displaystyle x_{j}=\operatorname {sign} (a_{ij})}$ ${\displaystyle j=1,\ldots ,n}$ ${\displaystyle \|\cdot \|_{1}}$ ${\displaystyle 1}$ ${\displaystyle \infty }$

Examples

Real matrix

The row sum norm of the real (2 × 3) matrix

${\displaystyle A={\begin{pmatrix}1&{-2}&{-3}\\2&3&{-1}\end{pmatrix}}}$

is calculated as

${\displaystyle \|A\|_{\infty }=\max\{|1|+|{-2}|+|{-3}|,|2|+|3|+|{-1}|\}=\max\{6,6\}=6}$.

Complex matrix

The row sum norm of the complex (2 × 3) matrix

${\displaystyle A={\begin{pmatrix}1&-2i&3-i\\2i&3&-1-i\end{pmatrix}}}$

is calculated as

${\displaystyle \|A\|_{\infty }=\max\{|1|+|-2i|+|3-i|,|2i|+|3|+|-1-i|\}=\max\{3+{\sqrt {10}},5+{\sqrt {2}}\}=5+{\sqrt {2}}}$.

properties

Standard properties

The norm axioms definiteness , absolute homogeneity and subadditivity follow for the row sum norm directly from the corresponding properties of natural matrix norms. In particular, the line sum norm is thus also sub-multiplicative and compatible with the maximum norm , that is, it applies

${\displaystyle \|A\cdot x\|_{\infty }\leq \|A\|_{\infty }\cdot \|x\|_{\infty }}$

for all matrices and all vectors and the row sum norm is the smallest norm with this property. ${\displaystyle A\in {\mathbb {K} }^{m\times n}}$ ${\displaystyle x\in {\mathbb {K} }^{n}}$

Adjoint

For an adjoint matrix (in the real case transposed matrix ) we have ${\displaystyle A^{H}\in {\mathbb {K} }^{n\times m}}$

${\displaystyle \|A^{H}\|_{\infty }=\max _{i=1,\ldots ,m}{\sum _{j=1}^{n}|{\bar {a}}_{ji}|}=\max _{j=1,\ldots ,m}{\sum _{i=1}^{n}|a_{ij}|}=\|A\|_{1}}$,

where the complex conjugate number is to with the same amount. The row sum norm of an adjoint or transposed matrix thus corresponds to the column sum norm of the output matrix. The spectral norm of a matrix can be estimated upwards as a geometric mean from the row and column sum norm. ${\displaystyle {\bar {a}}}$ ${\displaystyle a}$

literature

• Hans Rudolf Schwarz, Norbert Köckler: Numerical Mathematics . 8th edition. Vieweg & Teubner, 2011, ISBN 978-3-8348-1551-4 .

Web links

• Eric W. Weisstein : Maximum Absolute RowSum Norm . In: MathWorld (English).