A scalar that quantifies the sensitivity of the solution of a linear problem to relative perturbations in data; for an invertible matrix A it is commonly defined as cond(A)=||A||·||A^{-1}|| in a chosen operator norm.

Principle

Local linear sensitivity: the condition number bounds the worst-case amplification of relative input errors into relative solution errors under linear perturbations and a chosen norm.

Demonstration

For a diagonal matrix with entries 1 and 10^{-6} under the 2-norm the condition number is 10^6, indicating that relative errors in the right-hand side can be amplified up to six orders of magnitude in the computed solution.

Misapplication

Using the global condition number of a matrix to predict numerical error for a nonlinear problem without local linearization, or assuming it captures algorithmic stability independent of the solver's numerical method.

Consequence

A large condition number signals potential loss of significant digits and amplified errors; a moderate condition number means relative stability under small perturbations when combined with a numerically stable algorithm.

Reversal

Viewing conditioning versus stability: a well-conditioned problem can be solved poorly by an unstable algorithm, while an ill-conditioned problem can still be handled if one controls and quantifies propagated errors.

Boundary

Defined relative to a choice of operator norm and only meaningful for invertible linear operators; singular or rank-deficient operators have infinite condition number in this linear invertibility sense.

Semantic Tension

Confused with algorithmic stability: condition number is a property of the mathematical problem (data-to-solution map) while stability refers to the numerical method used to produce an approximate solution.

Synthesis

The condition number compresses the worst-case relative amplification of input perturbations by a linear operator, revealing when problem data require extra precision or regularization to obtain reliable solutions.