matrix condition numberMatrixConditionNumber2002-09-28 13:12:022006-10-07 12:27:04Definitionill-conditionedwell-conditioned\usepackage{amssymb}
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\providecommand{\norm}[1]{\lVert#1\rVert}\section{Matrix Condition Number}
The \emph{condition number for matrix inversion} with respect to a matrix norm
$\norm{\cdot}$ of a square matrix $A$ is defined by
\[
\kappa(A) = \Vert A \Vert \Vert A^{-1} \Vert\,,
\]
if $A$ is non-singular; and $\kappa(A) = +\infty$ if $A$ is singular.
The condition number is a measure of stability or sensitivity of a matrix (or the linear system it represents) to numerical operations. In other words, we may not be able to trust the results of computations on an ill-conditioned matrix.
Matrices with condition numbers near 1 are said to be \emph{well-conditioned}. Matrices with condition numbers much greater than one (such as around $10^5$ for a $5 \times 5$ Hilbert matrix) are said to be \emph{ill-conditioned}.
If $\kappa(A)$ is the condition number of $A$, then $\kappa(A)$ measures
a sort of inverse distance from $A$ to the set of singular matrices,
normalized by $\norm{A}$.
Precisely, if $A$ is invertible, and $\norm{B - A} < \norm{A^{-1}}^{-1}$,
then $B$ must also be invertible. On the other hand, in the case of the $2$-norm,
there always exists a singular matrix $B$ such that $\norm{B-A}_2 = \norm{A^{-1}}_2^{-1}$
(so the distance estimate is sharp).
\begin{thebibliography}{3}
\bibitem{Golub} Golub and Van Loan. \emph{Matrix Computations}, 3rd edition. Johns Hopkins University Press, 1996.
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