Column sum norm

Illustration of the column sum norm

The column sum norm is in the mathematics of the total standard derived natural matrix norm . The column sum norm of a matrix corresponds to the maximum amount sum of all its columns. It is sub-multiplicative and compatible with the sum norm . The column sum norm is used in particular in linear algebra and numerical mathematics .

definition

The column sum norm of a matrix with as the field of real or complex numbers is the natural matrix norm derived from the sum norm and is thus defined as ${\displaystyle \|\cdot \|_{1}}$ ${\displaystyle A\in {\mathbb {K} }^{m\times n}}$ ${\displaystyle \mathbb {K} }$

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

The column sum norm clearly corresponds to the greatest possible expansion factor that results from applying the matrix to a vector with sum sum one. The name-giving representation applies to the column sum norm

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

Here, the sum in the amount of strokes for solid was used for one of the unit vectors with maximum is. The calculation of the column total standard is therefore carried out by determining the total amount of each column and then by selecting the maximum of these values. The following rule of thumb helps to distinguish it from the related line sum norm : it is vertical and stands for the columns, while the one is horizontal and stands for the lines. ${\displaystyle i}$ ${\displaystyle x=\pm e_{j}}$ ${\displaystyle j=1,\ldots ,n}$ ${\displaystyle \|\cdot \|_{\infty }}$ ${\displaystyle 1}$ ${\displaystyle \infty }$

Examples

Real matrix

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

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

is calculated as

${\displaystyle \|A\|_{1}=\max\{|1|+|2|,|{-2}|+|3|,|{-3}|+|{-1}|\}=\max\{3,5,4\}=5}$.

Complex matrix

The column-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

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

properties

Standard properties

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

${\displaystyle \|A\cdot x\|_{1}\leq \|A\|_{1}\cdot \|x\|_{1}}$

for all matrices and all vectors and the column 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 ${\displaystyle A^{H}\in {\mathbb {K} }^{n\times m}}$

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

where the complex conjugate number is to with the same amount. The column sum norm of an adjoint or transposed matrix thus corresponds to the row 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 Column Sum Norm . In: MathWorld (English).