Condition Number

Updated 01 Oct 2026

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What it is
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  measures the sensitivity of a problem's solution to small changes in its inputs
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Formula
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  for nonsingular $A$ , $\kappa(A)=|A||A^{-1}|$
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  in the 2-norm, $\kappa_2(A)=\sigma_{\max}/\sigma_{\min}$
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Example
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  for $A=\begin{pmatrix}1&2 \\1.0001&2\end{pmatrix}$ , $\kappa_\infty(A)=60002$ .
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  changing $b=(3,3.0001)^T$ to $(3,3.0002)^T$ changes the solution from $(1,1)^T$ to $(2,0.5)^T$ .
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Interpretation
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  a small residual can coexist with a substantial solution error for an ill-conditioned system
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  conditioning belongs to the problem; numerical stability describes how an algorithm behaves
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Related
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  Matrix Norm
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  Preconditioning

Linked references 5