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5.54.6  The operator norm of a matrix: matrix_norm, l1norm, l2norm, norm, specnorm, linfnorm

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:

B := [[1,2,3],[3,-9,6],[4,5,6]]

then:

matrix_norm(B,1)

or:

l1norm(B)

or:

colNorm(B)

Output:

16

since max(1+3+4, 2+9+5, 3+6+6)=16.

Input:

matrix_norm(B,2)

or:

l2norm(B)

or:

max(SVL(B))

Output:

11.2449175989

Input:

matrix_norm(B,inf)

or:

linfnorm(B)

or:

rowNorm(B)

Output:

18

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:EF is continuous exactly when there is a number K such that ||f(x)||FK ||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:EF 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.


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