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).

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