Pseudo norm

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In algebra, a pseudonorm is a weakened variant of a norm in which the property of homogeneity is weakened to subhomogeneity. Just as the norm can be seen as a generalization of an amount into the multidimensional, the pseudo norm is related to a pseudo amount in which, in contrast to the amount, the condition of multiplicativity is weakened to submultiplicativity .

definition

Let be a - (left) module over a unitary ring with pseudo amount . A mapping into the non-negative real numbers is called a pseudonorm if all and the following properties apply: ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle (R,|\cdot |)}$ ${\displaystyle \|\cdot \|\colon M\to \mathbb {R} _{0}^{+}}$ ${\displaystyle a,b\in M}$ ${\displaystyle \lambda \in R}$

(1) (definiteness) ${\displaystyle \|a\|=0\;\Leftrightarrow \;a=0}$

(2) (subhomogeneity) ${\displaystyle \|\lambda a\|\leq |\lambda |\cdot \|a\|}$

(3) (triangle inequality). ${\displaystyle \|a+b\|\leq \|a\|+\|b\|}$

Will (2) tighten too

(2a) (homogeneity), ${\displaystyle \|\lambda a\|=|\lambda |\cdot \|a\|}$

that's the name of a norm . ${\displaystyle \|\cdot \|}$

The terminology is not clear in the literature; with some authors the pseudo amount is already referred to as the pseudo norm.

properties

• Is the norm pseudo even a norm on , as is necessarily the associated pseudo amount an amount on . ${\displaystyle \|\cdot \|}$ ${\displaystyle M}$ ${\displaystyle |\cdot |}$ ${\displaystyle R}$

p-pseudo norms

definition

If a unitary ring with a pseudo-amount, then on the module through ${\displaystyle (R,|\cdot |)}$ ${\displaystyle R}$ ${\displaystyle R^{n}}$

${\displaystyle \|v\|_{p}={\sqrt[{p}]{\sum _{i=1}^{n}|v_{i}|^{p}}}}$

for each or through ${\displaystyle p\in [1,\infty )}$

${\displaystyle \|v\|_{\infty }=\max _{i=1}^{n}|v_{i}|}$

for a pseudo norm that explains p-pseudo norm . ${\displaystyle p=\infty }$

comment

In order for this definition to be meaningful, the pseudonorm properties must be shown. The Minkowski inequality is used to prove the triangle inequality .

properties

• For true always . ${\displaystyle 1\leq p\leq q\leq \infty }$ ${\displaystyle \|v\|_{q}\leq \|v\|_{p}}$
• For true always . ${\displaystyle 1\leq p<\infty }$ ${\displaystyle \|v\|_{p}\leq {\sqrt[{p}]{n}}\,\|v\|_{\infty }}$

application

If a unitary ring with a pseudo amount, then we can consider the polynomial rings or and the matrix rings as modules. This is done by “writing” the coefficients one after the other. The pseudo norms can thus be explained by the above definition . These are generally not sub-multiplicative on the polynomial algebras and on the matrix algebras . The following special cases are all the more valuable: ${\displaystyle (R,|\cdot |)}$ ${\displaystyle R[X]}$ ${\displaystyle R[X_{1},\dots ,X_{n}]}$ ${\displaystyle R^{m\times n}}$ ${\displaystyle R}$ ${\displaystyle p}$

• The pseudonorm is submultiplicative on polynomial algebra . ${\displaystyle 1}$ ${\displaystyle R[X]}$
• For two multipliable matrices and as well as selected with applies ${\displaystyle A\in R^{l\times m}}$ ${\displaystyle B\in R^{m\times n}}$ ${\displaystyle p,q\in [1,\infty ]}$ ${\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}$

${\displaystyle \|AB\|_{p}\leq \|A\|_{p}\,\|B\|_{\min(p,q)}}$,

${\displaystyle \|AB\|_{p}\leq \|A\|_{\min(p,q)}\,\|B\|_{p}}$.

• The Hölder inequality and the Minkowski inequality are used to prove this statement .
• If , then the pseudonorm is submultiplicative for all multipliable matrices over , and this is especially true for the algebras of the square matrices. ${\displaystyle p\in [1,2]}$ ${\displaystyle p}$ ${\displaystyle R}$ ${\displaystyle R^{n\times n}}$
• Example for the pseudo norm: If R is a commutative ring with a pseudo amount and M is a matrix over R with the lines , then the weakened Hadamard inequality with the 1 pseudo norm applies . ${\displaystyle 1}$ ${\displaystyle n\times n}$ ${\displaystyle M_{1},\dots ,M_{n}}$ ${\displaystyle \textstyle |\det M|\leq \prod _{i=1}^{n}\|M_{i}\|_{1}}$

Applications and meaning

Associative algebras

On associative algebras , structures that have both norm and magnitude properties at the same time are relatively easy to classify: Let be an associative algebra over a commutative unitary ring with pseudo magnitude . ${\displaystyle A}$ ${\displaystyle R}$ ${\displaystyle (R,|\cdot |)}$

• If a submultiplicative pseudo norm is on as a module, then a pseudo amount is on as a ring. ${\displaystyle \|\cdot \|}$ ${\displaystyle A}$ ${\displaystyle \|\cdot \|}$ ${\displaystyle A}$
• If there is even a multiplicative pseudonorm, an amount is added . ${\displaystyle \|\cdot \|}$ ${\displaystyle \|\cdot \|}$ ${\displaystyle A}$

Iterative construction of polynomial and matrix algebras

A large number of important complexity estimates in computer algebra work for pseudonorms in matrix and polynomial algebras over rings with pseudo magnitude.

The following iterative construction of associative algebras such as polynomial and matrix algebras is often used to obtain such estimates:

Assuming a basic ring R with a pseudo amount (this can often still be a real amount in practice), let us give an associative R -algebra A with a submultiplicative pseudo norm. Then A is in particular itself a ring with a pseudo amount, over which modules, polynomial and matrix rings can be viewed. In this way, for example, the iterative construction of the polynomial algebras is possible, with each intermediate algebra itself being equipped with a pseudo norm. ${\displaystyle R[X_{1},\dots ,X_{n}]=R[X_{1},\dots ,X_{n-1}][X_{n}]}$

Example: Pseudodivision of polynomials in several variables

Let R be a commutative unitary ring and the polynomial algebra in n variables R . Then it is explained by a non-Archimedean pseudo amount on the polynomial ring. Let the total degree of f with the additional convention . The restriction of this pseudo amount to R results in the trivial pseudo amount, which is always 1 with the exception of zero, which is given the value 0. With regard to this pseudo amount on R , the amount is also a norm on , now understood as the R module. If R is also an integrity ring, there is even a non-Archimedean amount on the polynomial ring. With these tools one can derive a valuable estimate of the coefficient growth in the " pseudo division with remainder" with respect to one variable from polynomials in several variables. ${\displaystyle R[X_{1},\dots ,X_{n}]}$ ${\displaystyle |f|:=2^{\operatorname {grad} f}}$ ${\displaystyle \operatorname {grad} f}$ ${\displaystyle 2^{\operatorname {grad} (0)}=2^{-\infty }=0}$ ${\displaystyle f\mapsto 2^{\operatorname {grad} f}}$ ${\displaystyle R[X_{1},\dots ,X_{n}]}$ ${\displaystyle f\mapsto 2^{\operatorname {grad} f}}$

literature

• Jürgen Klose: Fast polynomial arithmetic for the exact solution of the Fermat-Weber problem . Ed .: Friedrich-Alexander University Erlangen-Nuremberg. July 1993. Here pp. 48-62.

Individual evidence

1. ↑ LA Bokhut ', IV L'vov, IR Shafarevich: Noncommutative Rings . In: AI Kostrikin, IR Shafarevich (ed.): Algebra II . Springer, Berlin 1991 (English).
2. ↑ George E. Collins, Ellis Horowitz: The Minimum Root Separation of a Polynomial . In: Mathematics of Computation . tape 28 , no. 126 , April 1974, p. 589-597 (English).