Overall norm

The overall standard is in the mathematics one on the maximum norm -based matrix norm . It is defined as the maximum amount of the matrix element multiplied by the geometric mean of the number of rows and columns in the matrix . The overall norm is sub-multiplicative and, with certain restrictions on the dimensions of the matrix, compatible with all p norms , but it is not an operator norm . It is used in particular in numerical linear algebra .

definition

The overall norm of a real or complex ( m × n ) matrix with as the field of real or complex numbers is defined as ${\displaystyle \|\cdot \|_{G}}$ ${\displaystyle A\in {\mathbb {K} }^{m\times n}}$ ${\displaystyle \mathbb {K} }$

${\displaystyle \|A\|_{G}:={\sqrt {mn}}\cdot \!\max _{{i=1,\ldots ,m} \atop {j=1,\ldots ,n}}|a_{ij}|}$,

thus the product of the geometric mean of the number of rows and columns in the matrix with the maximum of the amounts of all matrix elements . With the exception of the prefactor, the overall standard thus corresponds to the maximum entry of a vector of length in which all entries in the matrix are noted below one another, and thus to the maximum standard of this vector. ${\displaystyle a_{ij}}$ ${\displaystyle m\cdot n}$

For the special case of a square matrix , the overall norm is through ${\displaystyle A\in {\mathbb {K} }^{n\times n}}$

${\displaystyle \|A\|_{G}=n\cdot \!\max _{i,j=1,\ldots ,n}|a_{ij}|}$

given.

Examples

Real matrix

The overall norm of the real (2 × 2) matrix

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

is given as

${\displaystyle \|A\|_{G}=2\cdot \max _{i,j=1,2}|a_{ij}|=2\cdot \max\{|1|,|{-}1|,|{-}2|,|3|\}=2\cdot \max\{1,1,2,3\}=6}$.

Complex matrix

The overall norm of the complex (2 × 2) matrix

${\displaystyle A={\begin{pmatrix}1&i\\-2i&3-4i\\\end{pmatrix}}}$

is given as

${\displaystyle \|A\|_{G}=2\cdot \max _{i,j=1,2}|a_{ij}|=2\cdot \max\{|1|,|i|,|{-}2i|,|3-4i|\}=2\cdot \max\{1,1,2,5\}=10}$.

properties

Norm axioms

Since the sum of two matrices and the multiplication of a matrix by a scalar are defined component-wise, the norm properties of definiteness , absolute homogeneity and subadditivity follow directly from the corresponding properties of the maximum norm for vectors. The scaling with the constant prefactor has no influence on the statements. ${\displaystyle A,B\in {\mathbb {K} }^{m\times n}}$ ${\displaystyle {\sqrt {mn}}}$

Sub-multiplicativity

The overall norm is sub-multiplicative , i.e. for matrices and applies ${\displaystyle A\in {\mathbb {K} }^{m\times n}}$ ${\displaystyle B\in {\mathbb {K} }^{n\times l}}$

${\displaystyle \|A\,B\|_{G}\leq \|A\|_{G}\cdot \|B\|_{G}}$,

like with the help of the triangle inequality and with the estimation of a sum of matrix elements by the corresponding multiple of the maximum element over

${\displaystyle {\begin{aligned}\|A\,B\|_{G}&={\sqrt {ml}}\cdot \!\max _{{i=1,\ldots ,m} \atop {k=1,\ldots ,l}}\left|\sum _{j=1}^{n}a_{ij}b_{jk}\right|\leq {\sqrt {ml}}\cdot \!\max _{{i=1,\ldots ,m} \atop {k=1,\ldots ,l}}\sum _{j=1}^{n}|a_{ij}|\cdot |b_{jk}|\leq {\sqrt {ml}}\cdot n\cdot \!\max _{{i=1,\ldots ,m} \atop {j=1,\ldots ,n}}|a_{ij}|\cdot \!\max _{{j=1,\ldots ,n} \atop {k=1,\ldots ,l}}|b_{jk}|=\\&={\sqrt {mn}}\cdot \!\max _{{i=1,\ldots ,m} \atop {j=1,\ldots ,n}}|a_{ij}|\cdot {\sqrt {nl}}\cdot \!\max _{{j=1,\ldots ,n} \atop {k=1,\ldots ,l}}|b_{jk}|=\|A\|_{G}\cdot \|B\|_{G}\end{aligned}}}$

can be shown. This also explains the reason for the scaling, since the overall standard is generally not sub-multiplicative without this prefactor.

compatibility

The overall standard is compatible with all p standards , provided that applies to and to . Under these restrictions, the inequality holds for a matrix and a vector ${\displaystyle m\leq n}$ ${\displaystyle p<2}$ ${\displaystyle m\geq n}$ ${\displaystyle p>2}$ ${\displaystyle A\in {\mathbb {K} }^{m\times n}}$ ${\displaystyle x\in {\mathbb {K} }^{n}}$

${\displaystyle \|A\,x\|_{p}\leq \|A\|_{G}\cdot \|x\|_{p}}$.

The compatibility follows from the chain of inequalities

${\displaystyle {\begin{aligned}\|A\,x\|_{p}^{p}&=\sum _{i=1}^{m}\left|\sum _{j=1}^{n}a_{ij}x_{j}\right|^{p}\!\leq \sum _{i=1}^{m}\max _{j=1,\ldots ,n}|a_{ij}|^{p}\cdot \left|\sum _{j=1}^{n}x_{j}\right|^{p}\!\leq m\cdot \!\max _{{i=1,\ldots ,m} \atop {j=1,\ldots ,n}}|a_{ij}|^{p}\cdot \|x\|_{1}^{p}\leq \\&\leq {\frac {m}{({\sqrt {mn}}\,)^{p}}}\,\|A\|_{G}^{p}\cdot n^{p-1}\,\|x\|_{p}^{p}=\left({\frac {m}{n}}\right)^{1-p/2}\|A\|_{G}^{p}\cdot \|x\|_{p}^{p}\,,\end{aligned}}}$

where the prefactor is bounded by one under exactly the above conditions . The 1-norm was estimated by the p -norm and, as with submultiplicativity, the sum was replaced by the maximum and the triangular inequality was repeatedly applied. ${\displaystyle \|x\|_{1}\leq {\sqrt[{p}]{n^{p-1}}}\,\|x\|_{p}}$

The overall norm is therefore always compatible with the Euclidean norm . It is only compatible with the sum norm and all other p norms for if the number of rows is at most as large as that of the columns. It is only compatible with the maximum norm and all other p norms for if the number of rows is at least as large as that of columns. For square matrices the overall norm is compatible with all p norms. ${\displaystyle p<2}$ ${\displaystyle p>2}$

Cannot be represented as an operator norm

The overall standard is no operator norm and therefore no natural matrix norm , that is, there is no vector norm so ${\displaystyle \|\cdot \|,}$

${\displaystyle \max _{x\neq 0}{\frac {\|Ax\|}{\|x\|}}=\|A\|_{G}}$

applies, as each operator norm for the identity matrix the value one must have, but for greater results in a value less than one. If the overall norm for the identity matrix is ​​scaled to one, then the submultiplicativity, which is another property of every operator norm, is lost. ${\displaystyle I\in {\mathbb {K} }^{n\times n}}$ ${\displaystyle \|I\|_{G}=n}$ ${\displaystyle n\geq 2}$

literature

• Gene Golub , Charles van Loan: Matrix Computations . 3. Edition. Johns Hopkins University Press, 1996, ISBN 978-0-8018-5414-9 .
• Roger Horn, Charles R. Johnson: Matrix Analysis . Cambridge University Press, 1990, ISBN 978-0-521-38632-6 .
• Hans Rudolf Schwarz, Norbert Köckler: Numerical Mathematics . 8th edition. Vieweg & Teubner, 2011, ISBN 978-3-8348-1551-4 .