Math lecture for geophysicist
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Matrix Norm
Prepared byDr Amin E. Khalil
Definitions:
Matrix norm is a measure of the size of a matrix, or a distance between two matrices.
p
p
Xp X
AXA
0sup
How to compute matrex norm?
The solution of matrix norm involves the following two steps:1- find candidate for the norm, say K, that satisfy the relation:
ppXKAX
2- Find at one nonzero vector that satisfy
ppXKAX 00
then KAp
Matrix L1 norm
Matrix 1-norm is the maximum of the column sum. To investigate this let m×n matrix represented in the form
yielding
Which equals
The first step is thus obtained where K= max ||Aj||p
To maximize We choose X in the form
The matrix L1 norm is the column with largest sum, thus
31A
L∞ norm
What is the infinity norm of the previous example?
The answer is 3
L2 norm
Newton-Raphson method