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'matrix condition number'
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| Title of object: |
matrix condition number |
| Canonical Name: |
MatrixConditionNumber |
| Type: |
Definition |
| Created on: |
2002-09-28 13:12:02 |
| Modified on: |
2004-03-30 19:51:15 |
| Classification: |
msc:65F35, msc:15A12 |
| Defines: |
ill-conditioned, well-conditioned |
| Synonyms: |
matrix condition number=matrix condition number
matrix condition number=condition number |
Revision comment (for changes between this and next version):
| Changes for correction #6644 ('Condition number and distance to a set of singular matrices'). |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
%\usepackage{psfrag}
%\usepackage{graphicx}
%\usepackage{xypic} |
Content:
\section{Matrix Condition Number}
The \emph{matrix condition number} $\kappa(A)$ of a square matrix $A$ is defined as
$$ \kappa(A) = \Vert A \Vert \Vert A^{-1} \Vert $$
where $ \Vert \cdot \Vert $ is any valid matrix norm.
The condition number is basically 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 $\kappa_p(A)$ is the condition number of $A$ in the $p$-norm, then $\kappa_p(A)$ measures the relative $p$-norm distance from $A$ to the set of singular matrices.
\begin{thebibliography}{3}
\bibitem{Golub} Golub and Van Loan, \emph{Matrix Computations}, 3rd edition. Johns Hopkins University Press 1996
\end{thebibliography} |
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