The matrix_norm command takes two arguments,
a matrix A and a second argument of
either 1, 2 or inf.
matrix_norm returns the operator norm of the operator
associated to the matrix. (See the reminder below for a discussion of
operator norms.) The operator norm will be relative to the ℓ1,
ℓ2 or ℓ∞ norm on ℝn, depending on the second
argument.
Note that
Input:
then:
or:
or:
Output:
since max(1+3+4, 2+9+5, 3+6+6)=16.
Input:
or:
or:
Output:
Input:
or:
or:
Output:
since max(1+2+3, 3+9+6, 4+5+6)=18.
Reminder:
In mathematics, particularly functional analysis, a linear function
between two normed spaces f:E → F is continuous exactly when there
is a number K such that ||f(x)||F ≤ K ||x|| for all x in E.
For this reason, they are also called bounded linear functions. The
infimum of all such K is defined to be the operator norm of f, and
it depends on the norms of E and F. There are other
characterizations of the operator norm of f, such as the supremum of
||f(x)||F over all x in E with ||x||E ≤ 1.
If E and F are finite dimensional, then any linear function f:E→ F will be bounded.
Any m× n matrix A = (ajk) corresponds to a linear function f:ℝn → ℝm defined by f(x) = Ax. We will refer to the operator norm of f as the operator norm of A.
| ( |
| |ajk|, |
| ( |
| |ajk|, |