Condition number

From Encyclopedia of Mathematics
Jump to: navigation, search

The condition number of a square matrix $A$ is defined as \begin{equation} \kappa(A) = \|A\|_2\cdot\|A^{-1}\|_2, \end{equation} where $\|\cdot\|_2$ is the spectral norm, that is, the matrix norm induced by the Euclidean norm of vectors. If A is singular then $\kappa(A)=\infty$. In numerical analysis the condition number of a matrix $A$ is a way of describing how well or badly the system $Ax=b$ could be approximated. If $\kappa(A)$ is small the problem is well-conditioned and if $\kappa(A)$ is large the problem is rather ill-conditioned.

Another expression for the condition number is $\kappa(A) = \dfrac{\sigma_{\max}}{\sigma_{\min}}$, where $\sigma_{\max}$ and $\sigma_{\min}$ are the maximal and minimal singular values of matrix $A$. If $A$ is a symmetric matrix then $\kappa(A)=\left|\dfrac{\lambda_\max}{\lambda_\min}\right|$, where $\lambda_\max$ and $\lambda_\min$ denote the largest and smallest eigenvalues of $A$.

How to Cite This Entry:
Condition number. Encyclopedia of Mathematics. URL: