# matrix condition number

## 1 Matrix Condition Number

The condition number for matrix inversion with respect to a matrix norm $\lVert\cdot\rVert$ of a square matrix $A$ is defined by

 $\kappa(A)=\|A\|\|A^{-1}\|\,,$

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 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 $\lVert A\rVert$. Precisely, if $A$ is invertible, and $\lVert B-A\rVert<\lVert A^{-1}\rVert^{-1}$, 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 $\lVert B-A\rVert_{2}=\lVert A^{-1}\rVert_{2}^{-1}$ (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