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Homematrix condition number

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# 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.

## Mathematics Subject Classification

15A12*no label found*65F35

*no label found*

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