condition number of a matrix

How much a linear system A x = b can amplify a small relative error in b (or in A) into a relative error in the solution x -- the abstract condition number from Chapter 1, made concrete for exactly this one problem.

The condition number of a nonsingular matrix $A$ is $\kappa(A)=\lVert A\rVert\,\lVert A^{-1}\rVert\ge1$ , in a stated matrix norm. A small $\kappa(A)$ (near $1$ ) means $A\mathbf{x}=\mathbf{b}$ is well-conditioned; a large $\kappa(A)$ means it's ill-conditioned, and the relative-error bound $\lVert\mathbf{x}-\hat{\mathbf{x}}\rVert/\lVert\mathbf{x}\rVert\le\kappa(A)\,\lVert\mathbf{r}\rVert/\lVert\mathbf{b}\rVert$ holds for any approximation $\hat{\mathbf{x}}$ with residual $\mathbf{r}=\mathbf{b}-A\hat{\mathbf{x}}$ , regardless of which algorithm produced $\hat{\mathbf{x}}$ .

Examples

The identity matrix has $\kappa(I)=1$ exactly (perfectly conditioned); a Hilbert matrix's condition number grows explosively with its size, a classic ill-conditioned example.