Matrix norms and conditioning

Measures the maximum factor by which a matrix can amplify vectors under a given norm.

For $A=(a_{ij})\in {\mathbb{R}}^{m\times n}$

Extends the Euclidean norm to the set of all matrix entries.

It is the maximum absolute column sum.

It is the maximum absolute row sum.

It is the largest Euclidean stretching factor produced by a matrix.

For a submultiplicative matrix norm

The amplification of a composition does not exceed the product of the maximum amplifications of each transformation.

For an invertible matrix and a compatible norm

Measures the sensitivity of a linear problem to small perturbations in the data.

• The formula \|A\|\|A^{-1}\| requires A to be invertible.
• For a singular matrix, the condition number is considered infinite in this context.