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=(a_jk) corresponds to a linear function f:ℝⁿ → ℝ^m defined by f(x)=Ax. The operator norm of A will be the operator norm of f.

• If ℝⁿ and ℝ^m both have the ℓ₁ norm, namely for x=(x₁,x₂,…) the norm is ||x||= ∑_j |x_j|, 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 ℝ^m both have the ℓ₂ norm, namely for x=(x₁,x₂,…) the norm is ||x||= √∑_j x_j² (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 ℝ^m both have the ℓ_∞ norm, namely for x=(x₁,x₂,…) the norm is |x|= max_j |x_j|, 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

then:

or:

or:

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

or:

or:

or:

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