Skip to content

Condition Number

Formula

\[ \kappa(A) = \|A\|\,\|A^{-1}\| \]

Parameters

  • \(A\): nonsingular matrix
  • \(\|\cdot\|\): matrix norm (often 2-norm)

What it means

Measures sensitivity of linear systems \(Ax=b\) to perturbations.

What it's used for

  • Predicting sensitivity of linear solves to noise.
  • Comparing numerical stability of matrices.

Key properties

  • \(\kappa(A)\ge 1\)
  • \(\kappa_2(A)=\sigma_{\max}/\sigma_{\min}\)

Common gotchas

  • Large \(\kappa\) implies numerical instability.
  • Depends on the chosen norm.

Example

For \(A=I\), \(\kappa(A)=1\). For \(A=\operatorname{diag}(1,0.01)\), \(\kappa=100\).

How to Compute (Pseudocode)

Input: nonsingular matrix A and chosen norm
Output: condition number kappa(A)

if using 2-norm:
  compute singular values of A
  return sigma_max / sigma_min
else:
  compute ||A|| and ||A^{-1}|| (or estimate them)
  return ||A|| * ||A^{-1}||

Complexity

  • Time: Depends on the norm and method; exact 2-norm condition numbers typically use SVD/eigensolver computations, while practical solvers often estimate conditioning more cheaply
  • Space: Depends on matrix storage and decomposition/estimation workspaces
  • Assumptions: Dense exact computation can be expensive; norm estimators are common in practice for large systems

See also