1.2 Norms and Condition Number

The magnitude of any scalar number is easily measured by its modulus, which is non-negative, i.e. a for a ? . The same can be done for vectors in ⁿ and matrices in ^{m n}, through a non-negative measure called the norm. The following definition, using three norm axioms, determines if any such non-negative measure is a norm.

Definition 1.2.2. Let V be either ⁿ (for vectors) or ^{m n} (for matrices). A measure u of u ? V, satisfying the following Norm axioms, is a valid norm:

• u ? 0 for any u and u = 0 is and only if u = 0,

• ?u= ? u for any u and any ? ? ,

• u+ v ? u+ v for any u, v ? V.

Remark that the same axioms are also used for function norms.

One can verify that the following are valid vector norms [80], for x ? ⁿ,

(1.7)

and similarly the following are valid matrix norms, for A ? ^{m n},

(1.8)

where sup denotes supremum and note Ax ? ^m.

While the formulae for vector norms are easy, those for matrices are not. We need to take some specific p in order to present computable formulae from (1.8)

(1.9)

where A ? is partitioned in columns first A