Natural matrix norm

A natural matrix norm , induced matrix norm or limit norm is in mathematics a matrix norm derived from a vector norm as an operator norm . A natural matrix norm clearly corresponds to the greatest possible stretching factor that results from the application of the matrix to a vector. Natural matrix norms are always sub-multiplicative and compatible with the vector norm from which they were derived . They are even the smallest of all matrix norms compatible with this vector norm. Important natural matrix norms are the row sum norm , the spectral norm and the column sum norm . Natural matrix norms are used in particular in linear algebra and numerical mathematics .

definition

A matrix norm is called induced by a vector norm or a natural matrix norm if it is derived from it as an operator norm . The natural matrix norm of a real or complex matrix is thus defined as ${\displaystyle \|\cdot \|}$ ${\displaystyle \|\cdot \|_{V}}$ ${\displaystyle A\in {\mathbb {K} }^{m\times n}}$

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

where the norm in the numerator has a vector as an argument and the norm in the denominator has a vector as an argument . Since it to each vector with one on one are normalized vector, any natural matrix standard also has the representation ${\displaystyle Ax\in {\mathbb {K} }^{m}}$ ${\displaystyle x\in {\mathbb {K} }^{n}}$ ${\displaystyle x\neq 0}$ ${\displaystyle {\bar {x}}:=x/\|x\|_{V}}$

${\displaystyle \|A\|=\max _{\|x\|_{V}=1}\|Ax\|_{V}}$,

it is therefore sufficient to consider the maximum over all unit vectors . The natural matrix norm thus clearly corresponds to the greatest possible stretching factor that results from the application of the matrix to a unit vector. An equivalent definition of the natural matrix norm is

${\displaystyle \|A\|=\min \left\{r\geq 0\mid \|Ax\|_{V}\leq r\,\|x\|_{V}\;\forall \;x\neq 0\right\}}$

or analogous to it

${\displaystyle \|A\|=\min \left\{r\geq 0\mid \|Ax\|_{V}\leq r\;\forall \;\|x\|_{V}=1\right\}}$,

i.e. the radius of the smallest standard sphere that includes the set . ${\displaystyle r}$ ${\displaystyle \{Ax\mid \|x\|_{V}=1\}}$

example

Illustration of the matrix norm induced by the Euclidean norm

Find the matrix norm of the (2 × 2) matrix induced by the Euclidean vector norm ${\displaystyle \|\cdot \|_{2}}$

${\displaystyle A=\left({\begin{matrix}{\frac {1}{2}}&{\frac {3}{2}}\\-{\frac {1}{2}}&{\frac {3}{2}}\end{matrix}}\right)}$.

As a linear mapping, this matrix describes simultaneous stretching in -direction, compression in -direction and rotation by 45 °. In the picture opposite, the red circle corresponds to the unit circle in the Euclidean norm, i.e. the set of vectors with length one. The green ellipse is then the unit circle after transformation ( rotation extension ) through the matrix . The natural matrix norm of then corresponds to the length of that vector on the green ellipse whose length is maximum. In the example these are the two vectors ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle A}$ ${\displaystyle A}$

${\displaystyle \left(\,{\tfrac {3}{2}},\,{\tfrac {3}{2}}\,\right)}$   and   . ${\displaystyle \left(\,{-}{\tfrac {3}{2}},\,{-}{\tfrac {3}{2}}\,\right)}$

The natural matrix norm of with respect to the Euclidean norm is then the length of one of these vectors and thus ${\displaystyle A}$

${\displaystyle \|A\|=\max _{\|x\|_{2}=1}\|Ax\|_{2}={\sqrt {\left({\tfrac {3}{2}}\right)^{2}+\left({\tfrac {3}{2}}\right)^{2}}}=3{\tfrac {\sqrt {2}}{2}}\approx 2{,}12}$.

The blue circle is the circle with the smallest radius that encompasses the green set; its radius just corresponds to the natural matrix norm.

properties

In the following, the addition is omitted from the vector norm, since the argument of the norm implicitly makes it clear whether it is a matrix or a vector norm. ${\displaystyle V}$

Norm axioms

Every natural matrix norm satisfies the three norm axioms . The definiteness follows for from ${\displaystyle A\in {\mathbb {K} }^{m\times n}}$

${\displaystyle \|A\|=0\;\Leftrightarrow \;\max _{x\neq 0}{\frac {\|Ax\|}{\|x\|}}=0\;\Rightarrow \;\|Ax\|=0\;\forall \;x\neq 0\;\Rightarrow \;A=0}$.

The absolute homogeneity follows for and from the homogeneity of the vector norm ${\displaystyle A\in {\mathbb {K} }^{m\times n}}$ ${\displaystyle \alpha \in \mathbb {K} }$

${\displaystyle \|\alpha \cdot A\|=\max _{\|x\|=1}\|\alpha \cdot Ax\|=\max _{\|x\|=1}|\alpha |\cdot \|Ax\|=|\alpha |\cdot \max _{\|x\|=1}\|Ax\|=|\alpha |\cdot \|A\|}$.

The subadditivity also follows from the subadditivity of the vector norm ${\displaystyle A,B\in {\mathbb {K} }^{m\times n}}$

${\displaystyle \|A+B\|=\max _{\|x\|=1}\|Ax+Bx\|\leq \max _{\|x\|=1}\|Ax\|+\|Bx\|\leq \max _{\|x\|=1}\|Ax\|+\max _{\|x\|=1}\|Bx\|=\|A\|+\|B\|}$,

where the maximum of the sum was also estimated upwards by the sum of the maxima.

compatibility

Every natural matrix norm is compatible with the vector norm from which it was derived, i.e. for and applies ${\displaystyle A\in {\mathbb {K} }^{m\times n}}$ ${\displaystyle x\in {\mathbb {K} }^{n}}$

${\displaystyle \|Ax\|\leq \|A\|\cdot \|x\|}$,

which is straight from the definition of using as a minimal number ${\displaystyle \|A\|}$ ${\displaystyle r}$

${\displaystyle \|Ax\|\leq r\cdot \|x\|}$

follows. The natural matrix norm is thus even the smallest matrix norm that is compatible with the underlying vector norm. It will therefore limits norm or lub standard (after English. Lowest upper bound called). Furthermore, it follows from compatibility that every natural matrix norm of a square matrix is at least as large as its spectral radius .

Sub-multiplicativity

Every natural matrix norm is also sub-multiplicative , that is, for and applies ${\displaystyle A\in {\mathbb {K} }^{m\times n}}$ ${\displaystyle B\in {\mathbb {K} }^{n\times l}}$

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

what follows directly from the compatibility:

${\displaystyle \|A\,\cdot \,B\|=\max _{\|x\|=1}\|A\cdot B\cdot x\|\leq \max _{\|x\|=1}\|A\|\cdot \|B\cdot x\|=\|A\|\cdot \max _{\|x\|=1}\|B\cdot x\|=\|A\|\cdot \|B\|}$.

Special cases

Identity matrix

For the identity matrix , every natural matrix norm gives the value one, because it holds ${\displaystyle I}$

${\displaystyle \|I\|=\max _{\|x\|=1}\|Ix\|=\max _{\|x\|=1}\|x\|=1}$.

Inverse

If a square matrix is regular , then the natural matrix norm of its inverse holds ${\displaystyle A\in {\mathbb {K} }^{n\times n}}$

${\displaystyle \|A^{-1}\|=\max _{\|x\|=1}\|A^{-1}x\|=\max _{\|Ay\|=1}\|y\|=\left(\min _{\|Ay\|=1}\|y\|^{-1}\right)^{-1}=\left(\min _{\|x\|=1}\|Ax\|\right)^{-1}}$,

where the last equation results from the substitution . The natural matrix norm of the inverse is thus the reciprocal of the smallest stretching factor that results from the application of the matrix to a unit vector. This allows the condition of a regular matrix ${\displaystyle z=y\cdot \|Ax\|}$

${\displaystyle \kappa (A)=\|A\|\cdot \|A^{-1}\|}$

regarding a natural matrix norm as the ratio of the largest and smallest stretching factor that the matrix generates.

Examples of natural matrix norms

The most important natural matrix norms are induced by the p norms . Three of these natural matrix norms have their own names and special meanings.

• The column sum norm is represented by the sum norm induced norm . It corresponds to the maximum amount total of all columns in the matrix.
  ${\displaystyle \|A\|_{1}=\max _{\|x\|_{1}=1}\|Ax\|_{1}=\max _{j=1,\ldots ,n}\sum _{i=1}^{m}|a_{ij}|}$
  
• The spectral norm is represented by the Euclidean norm induced norm . It corresponds to the square root of the greatest eigenvalue of , where the adjoint matrix (in the real case transposed matrix ) is to.
  ${\displaystyle \|A\|_{2}=\max _{\|x\|_{2}=1}\|Ax\|_{2}={\sqrt {\lambda _{\max }(A^{H}A)}}}$
  ${\displaystyle A^{H}A}$ ${\displaystyle A^{H}}$ ${\displaystyle A}$
• The row sum norm is represented by the maximum norm induced norm . It corresponds to the maximum amount total of all rows in the matrix.
  ${\displaystyle \|A\|_{\infty }=\max _{\|x\|_{\infty }=1}\|Ax\|_{\infty }=\max _{i=1,\ldots ,m}{\sum _{j=1}^{n}|a_{ij}|}}$
  

generalization

Illustration of the natural matrix norm in two dimensions ${\displaystyle \|A\|_{\infty ,2}}$

More generally, a natural matrix norm can also be derived from two different vector norms, one norm measuring the size of a vector in the starting space and the other measuring the size of a vector in the target space. The matrix norm induced by these two norms is thus defined as ${\displaystyle \|\cdot \|\colon {\mathbb {K} }^{m\times n}\rightarrow \mathbb {R} _{+}}$ ${\displaystyle \|\cdot \|_{a}\colon {\mathbb {K} }^{n}\rightarrow \mathbb {R} _{+}}$ ${\displaystyle \|\cdot \|_{b}\colon {\mathbb {K} }^{m}\rightarrow \mathbb {R} _{+}}$

${\displaystyle \|A\|_{b,a}:=\max _{x\neq 0}{\frac {\|Ax\|_{b}}{\|x\|_{a}}}=\min \left\{r\geq 0\mid \|Ax\|_{b}\leq r\,\|x\|_{a}\;\forall \;x\neq 0\right\}}$.

Due to its definition as a minimum, it is compatible with the two vector norms in the sense of

${\displaystyle \|Ax\|_{b}\leq \|A\|_{b,a}\cdot \|x\|_{a}}$

and for submultiplicative with as the third vector norm in the sense of ${\displaystyle B\in {\mathbb {K} }^{n\times l}}$ ${\displaystyle \|\cdot \|_{c}\colon {\mathbb {K} }^{l}\rightarrow \mathbb {R} _{+}}$

${\displaystyle \|AB\|_{b,c}\leq \|A\|_{b,a}\cdot \|B\|_{a,c}}$,

as analogous to above due to compatibility

${\displaystyle \|AB\|_{b,c}=\max _{\|x\|_{c}=1}\|ABx\|_{b}\leq \max _{\|x\|_{c}=1}\|A\|_{b,a}\|Bx\|_{a}=\|A\|_{b,a}\max _{\|x\|_{c}=1}\|Bx\|_{a}=\|A\|_{b,a}\|B\|_{a,c}}$

applies. In practice, however, the same norm is used in the respective vector space instead of different vector norms.

literature

• Alfio Quarteroni , Riccardo Sacco, Fausto Saleri: Numerical Mathematics 1 . Springer, 2002, ISBN 3-540-67878-6 .
• Hans Rudolf Schwarz, Norbert Köckler: Numerical Mathematics . 8th edition. Vieweg & Teubner, 2011, ISBN 978-3-8348-1551-4 .
• Peter Knabner , Wolf Barth : Lineare Algebra. Basics and Applications. Springer Spectrum, Berlin / Heidelberg 2011, ISBN 978-3-642-32185-6 .

Web links

• Eric W. Weisstein : Natural Norm . In: MathWorld (English).
• Cam McLeman, Logan Hanks, Pedro Sanchez: Matrix p-norm . In: PlanetMath . (English)

Individual evidence

1. ↑ Schwarz, Köckler: Numerical Mathematics . S. 50 .