A scalar quantifying sensitivity of the solution of a problem to perturbations in input; for an invertible matrix A and a chosen norm, the matrix condition number is κ(A) = ||A|| · ||A^{-1}|| and measures worst-case relative amplification of errors.

Principle

The condition number compares the maximal output effect to minimal output effect under perturbations, capturing ill-conditioning when small input changes produce large relative output changes.

Demonstration

Solving Ax = b: if κ(A) is large then a small relative perturbation in b or small rounding errors can produce much larger relative error in the computed x; for diagonal matrix with entries 1 and 10^{-6}, κ ≈ 10^{6}.

Misapplication

Equating a small condition number with guaranteed small forward error regardless of algorithm; algorithmic numerical stability and rounding behavior also determine actual error.

Consequence

Condition numbers guide selection of preconditioners, regularization strength and algorithm choice; they predict (up to algorithmic stability) expected error magnification and inform problem reformulation.

Reversal

The reciprocal 1/κ measures relative numerical robustness (well-conditioned); reversing perspective focuses on problems where small input uncertainty cannot change outputs significantly.

Boundary

Defined for invertible linear operators with respect to a chosen norm; for singular or nonlinear problems one uses generalized condition measures or local linearization; different norms yield different numerical values.

Semantic Tension

Related but distinct from numerical stability: condition number is a property of the mathematical problem, whereas stability is a property of the algorithm used to solve it.

Synthesis

The condition number is the norm-based scalar that quantifies how input perturbations are amplified by a problem's mapping, signaling ill-posedness for large κ and motivating preconditioning or reformulation.