matrix condition number
1 Matrix Condition Number
The condition number^{} for matrix inversion^{} with respect to a matrix norm^{} $\parallel \cdot \parallel $ of a square matrix^{} $A$ is defined by
$$\kappa (A)=\parallel A\parallel \parallel {A}^{-1}\parallel ,$$ |
if $A$ is non-singular; and $\kappa (A)=+\mathrm{\infty}$ 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 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 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 $\parallel A\parallel $. Precisely, if $A$ is invertible, and $$, 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 ${\parallel B-A\parallel}_{2}={\parallel {A}^{-1}\parallel}_{2}^{-1}$ (so the distance estimate is sharp).
References
- 1 Golub and Van Loan. Matrix Computations, 3rd edition. Johns Hopkins University Press, 1996.
Title | matrix condition number |
Canonical name | MatrixConditionNumber |
Date of creation | 2013-03-22 13:04:17 |
Last modified on | 2013-03-22 13:04:17 |
Owner | stevecheng (10074) |
Last modified by | stevecheng (10074) |
Numerical id | 10 |
Author | stevecheng (10074) |
Entry type | Definition |
Classification | msc 15A12 |
Classification | msc 65F35 |
Synonym | matrix condition number |
Synonym | condition number |
Related topic | PropertyOfMatrixConditionNumber |
Defines | ill-conditioned |
Defines | well-conditioned |