##  [Condition Number](/condition-number-2) 

 Definition

A numerical measure of how sensitively the output of a function or solution of a linear system responds to small relative perturbations of the input; for invertible matrices under a chosen norm it is typically ||A|| ||A^{-1}||, and for matrices equals the ratio of largest to smallest singular value in the spectral norm.

 

 

 

 

 

 





## Principle

Principle

The condition number quantifies worst-case relative amplification of input error into output error for the problem and norm specified; it separates problem conditioning from algorithmic stability and depends on both operator and norm choice.

 

 

 

 

 





## Demonstration

Demonstration

For a 2×2 diagonal matrix diag(100,1) under spectral norm the condition number is 100/1=100, meaning a 1% relative perturbation in input can cause up to ~100% relative change in the solution. For a singular matrix the condition number is infinite, signaling no stable inverse.

 

 

 

 

## Misapplication

Misapplication

Omitting to specify the norm when quoting a condition number, or conflating a high condition number with algorithmic failure regardless of solver design. Using the matrix 2-norm condition number to predict errors for an algorithm whose stability is governed by a different norm may mislead.

 

 

 

 

 





## Consequence

Consequence

A large condition number indicates ill-conditioning: solutions are highly sensitive and require regularization, higher precision, or reformulation; a small condition number (near 1) indicates a well-conditioned problem where forward errors remain commensurate with input errors.

 

 

 

 

## Reversal

Reversal

Interpreting a low condition number as guarantee of accurate computed solution regardless of algorithmic behavior; the inverse statement is that good conditioning does not replace the need for numerically stable algorithms.

 

 

 

 

 





## Boundary

Boundary

Defined relative to a specified norm and problem (matrix inversion, linear solve, function evaluation). For nonlinear problems a local condition number uses the Jacobian; for noninvertible operators the classical condition number is infinite and must be replaced by condition measures on subspaces or regularized variants.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension arises between problem conditioning (intrinsic sensitivity measured by condition number) and numerical stability (algorithmic error amplification): a well-conditioned problem can be solved badly, and an ill-conditioned problem may be handled by specialized regularization or stabilized algorithms.

 

 

 

 

 





## Synthesis

Synthesis

The condition number is a norm-dependent scalar that measures worst-case relative amplification of input perturbations by a problem; for matrices it equals ||A||·||A^{-1}|| and, in the spectral norm, the ratio of largest to smallest singular value, guiding expectations about sensitivity and the need for stabilization.