### 15.4.6 Operator norm of a matrix

##### Operator norms.

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. (See Section 14.1.1 for norms on ℝn.) 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. The operator norm of A will be the operator norm of f.

- If ℝn and ℝm both have the ℓ1 norm, namely for x=(x1,x2,…) the norm is ||x||= ∑j |xj|, the operator norm of A is

  max
  k

  ∑
  j

  |ajk|.

  This is the column norm given by colnorm(A) (see Section 15.4.5).
- If ℝn and ℝm both have the ℓ2 norm, namely for x=(x1,x2,…) the norm is ||x||= √∑j xj2 (the usual Euclidean norm), the operator norm of A is the largest eigenvalue of f∗∘ f, where f∗ is the transpose of f, and so the largest singular value of f. This is given by l2norm (see Section 15.4.2) or max(SVL(A)) (see Section 15.3.7).
- If ℝn and ℝm both have the ℓ∞ norm, namely for x=(x1,x2,…) the norm is |x|= maxj |xj|, the operator norm of A is

  max
  j

  ∑
  k

  |ajk|.

  This is given by rownorm(A) (see Section 15.4.4).

##### Computing operator norms.

The matrix_norm command is a command which finds any of the above operator norms.

- matrix_norm takes two arguments:
  - A, a matrix.
  - arg, which can be 1, 2 or inf.
- matrix_norm(A,arg) returns an operator norm of the operator associated to the matrix, the norm is determined by arg.
  - If arg is 1, it is based on the ℓ1 norm on ℝn.

    matrix_norm(A,1) is the same as colnorm(A) and l1norm(A).

  - If arg is 2, it is based on the ℓ2 norm on ℝn.

    matrix_norm(A,2) is the same as l2norm(A) and norm(A).

  - If arg is inf, it is based on the ℓ∞ norm on ℝn.

    matrix_norm(A,inf) is the same as rownorm(A) and linfnorm(A).

#### Examples

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

then:

matrix_norm(B,1)

or:

l1norm(B)

or:

colnorm(B)

16 |  |  |  |  |  |  |  |  |  |

since max{1+3+4, 2+9+5, 3+6+6}=16.

matrix_norm(B,2)

or:

l2norm(B)

11.2449175989 |  |  |  |  |  |  |  |  |  |

matrix_norm(B,inf)

or:

linfnorm(B)

or:

rowNorm(B)

18 |  |  |  |  |  |  |  |  |  |

Indeed, max{1+2+3, 3+9+6, 4+5+6}=18.