Numerical Analysis/Matrix norm

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Definitions[edit]

The term Norm is often used without additional qualification to refer to a particular type of norm such as a Matrix norm or a Vector norm. Most commonly the unqualified term Norm refers to flavor of Vector norm technically known as the L2 norm. This norm is variously denoted , , or and give the length of an n-vector

Norms provide vector spaces and their linear operators with measures of size, length and distance more general than those we already use routinely in everyday life.

Induced Norm[edit]

If vector norms on Km and Kn are given (K is field of real or complex numbers), then one defines the corresponding induced norm or operator norm on the space of m-by-n matrices as the following maxima:

If m = n and one uses the same norm on the domain and the range, then the induced operator norm is a sub-multiplicative matrix norm.

The operator norm corresponding to the p-norm for vectors is:

In the case of and , the norms can be computed as:

which is simply the maximum absolute column sum of the matrix.
which is simply the maximum absolute row sum of the matrix.

Theorem: Induced Norms are really norms[edit]

If  is a vector norm on  then  is a matrix norm.

Theorem: Induced norms are submultiplicative[edit]

All  induced norms are sub-multiplicative.

Derivation of formula[edit]

If  is an  matrix, then 

Example computing [edit]

If

find .

Equivalence Of Norms[edit]

Equivalence Of Norms is defined as:

For any two norms ||·||α and ||·||β, we have

for some positive numbers r and s, for all matrices A in .  

This is true because the vector space  has the finite dimension .

Examples of matrix norm equivalence[edit]

For matrix  the following inequalities hold

  • , where is the rank of
  • , where is the rank of

Here, ||·||p refers to the matrix norm induced by the vector p-norm.

Example[edit]

We will show some of these norm equivalences for the matrix

Reference[edit]

  • Numerical Analysis by Richard L. Burden and J. Douglas Faires (EIGHT EDITION)
  • Elementary Numerical Analysis by Kendall Atkinson (Second Edition)
  • Applied Numerical Analysis by Gerald / Wheatley (Sixth Edition)
  • Theory and Problems of Numerical Analysis by Francis Scheid