User:Nikita2/sandbox2
Condition number of square matrix is defined as \begin{equation} \kappa(A) = \|A\|_2\cdot\|A^{-1}\|_2. \end{equation} If A is singular matrix ( degenerate matrix ) \kappa(A)=\infty. Condition number of 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.
Nikita2/sandbox2. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nikita2/sandbox2&oldid=29461