A scalar quantity that measures how relative perturbations in the input data of a problem are amplified into relative perturbations in its solution; for a linear system Ax = b with invertible A and a chosen norm, the matrix condition number is ||A||·||A^{-1}||.

Principle

Relative error propagation: first-order relative changes in output are bounded proportionally to the condition number times relative changes in input, so the condition number organizes sensitivity to data perturbations independent of a particular algorithm.

Demonstration

Solving Ax = b where A is nearly singular yields large changes in x for small changes in b; for example, Hilbert-like matrices have rapidly growing ||A||·||A^{-1}||, indicating loss of significant digits in computed solutions.

Misapplication

Using the condition number of a coefficient matrix as a definitive predictor of algorithmic accuracy without accounting for the chosen norm, problem formulation (e.g., scaling), or the numerical stability of the algorithm is incorrect.

Consequence

A large condition number warns that high relative errors or loss of precision are possible and motivates preconditioning, regularization, or reformulation to improve reliability of computed solutions.

Reversal

A problem with condition number near one is well-conditioned: small input perturbations lead to proportionally small output changes and standard numerical methods can typically preserve accuracy.

Boundary

Condition number is a property of the mathematical problem (with respect to a norm and solution mapping), not of a particular implementation; it becomes infinite for noninvertible linear maps and depends on norm choice.

Semantic Tension

Often conflated with numerical stability (an algorithm property); condition number quantifies the problem’s inherent sensitivity, whereas stability describes how an algorithm amplifies its own errors.

Synthesis

The condition number concisely characterizes intrinsic sensitivity: given a normed problem and solution map, it bounds relative output error from relative input perturbations and thus guides expectations and remedies (scaling, preconditioning, regularization).