Matrix Norms

Let . A norm of is denoted by .

Definitions

Suppose for all the definitions below.

1-norm

Maximum of the absolute column sums.

2-norm

Square root of the sum of all elements squared. Aka. Euclidean norm, or Frobenius norm. Defined for non-square matrices as well.

Infinity norm

Maximum of the absolute row sums.

Note

For any matrix :

Vector norm

Norm defined for column vectors.

Induced norm

Aka. operator norm, subordinate norm.

Suppose . The induced norm is defined for with respect to a given norm, .

Properties of Norms

Works for all types of norms.

Suppose are ordered.

4. (triangle inequality)

Unit Ball

A unit ball in with respect to a norm .

Unit disc

When , unit ball is also called the unit disc.