Numerical Analysis/Matrix norm
Definitions
editThe 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
editIf 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
editIf is a vector norm on then is a matrix norm.
Proof:
To show is a matrix norm we need to show several things.
if and only if .
If then for all vectors x with and so .
If then for all . Using , and successively implies that each column of is zero. Thus, if and only if
for scalars
Using the definition of Induced norms and the properties of the vector norm we have,
Again using the definition of Induced norms and the triangle inequality for the vector norm, we have
Theorem: Induced norms are submultiplicative
editAll induced norms are sub-multiplicative.
Proof:
We want to show .
We have
- .
Derivation of A∞ formula
editIf is an matrix, then
Proof::
First we show that
-
(
)
Let be an n-dimensional vector with Since is also an n-dimensional vector,
But so
and we have shown (1).
Now we will show the opposite inequality, that
-
(
)
Let p be an integer with
and be the vector with components
Then and for all so
and we have shown (2). Together, (1) and (2) yield
Example computing A∞
editIf
Solution:
and
so,
Equivalence Of Norms
editEquivalence 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
editFor 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
editWe will show some of these norm equivalences for the matrix
Solution:
First we compute several norms:
- Where r is the rank of the matrix.
We can then verify the norm equivalence
with our numbers
and
with our numbers
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