Difference between revisions of "Condition number"
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Condition number of a square matrix $A$ is defined as | Condition number of a square matrix $A$ is defined as | ||
\begin{equation} | \begin{equation} | ||
| − | \kappa(A) = \|A\|_2\cdot\|A^{-1}\|_2 | + | \kappa(A) = \|A\|_2\cdot\|A^{-1}\|_2, |
\end{equation} | \end{equation} | ||
| − | If A is [[Degenerate_matrix|singular]] then $\kappa(A)=\infty$. In [[numerical analysis]] the condition number of a matrix $A$ is a way of describing how well or bad 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. | + | where $\|\cdot\|_2$ is the spectral norm, that is, the matrix [[norm]] induced by the Euclidean norm of vectors. If A is [[Degenerate_matrix|singular]] then $\kappa(A)=\infty$. In [[numerical analysis]] the condition number of a matrix $A$ is a way of describing how well or bad 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 condition number is $\kappa(A) = \dfrac{\sigma_{\max}}{\sigma_{\min}}$, were $\sigma_{\max}$ and $\sigma_{\min}$ are maximal and minimal singular values of matrix $A$. If $A$ is a symmetric matrix then $\kappa(A)=\left|\dfrac{\lambda_\max}{\lambda_\min}\right|$, here $\lambda_\max$ and $\lambda_\min$ denote the largest and smallest eigenvalues of $A$. | Another expression for condition number is $\kappa(A) = \dfrac{\sigma_{\max}}{\sigma_{\min}}$, were $\sigma_{\max}$ and $\sigma_{\min}$ are maximal and minimal singular values of matrix $A$. If $A$ is a symmetric matrix then $\kappa(A)=\left|\dfrac{\lambda_\max}{\lambda_\min}\right|$, here $\lambda_\max$ and $\lambda_\min$ denote the largest and smallest eigenvalues of $A$. | ||
Revision as of 16:46, 22 February 2013
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 bad 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 condition number is $\kappa(A) = \dfrac{\sigma_{\max}}{\sigma_{\min}}$, were $\sigma_{\max}$ and $\sigma_{\min}$ are maximal and minimal singular values of matrix $A$. If $A$ is a symmetric matrix then $\kappa(A)=\left|\dfrac{\lambda_\max}{\lambda_\min}\right|$, here $\lambda_\max$ and $\lambda_\min$ denote the largest and smallest eigenvalues of $A$.
Condition number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Condition_number&oldid=29466