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 $\kappa (A)$ measuresa 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).