# Matrix condition number and reordering

Does the condition number change when a matrix is reordered by e.g. Cuthill Mckee or some other method?

## 1 Answer

No. A reordering of a matrix $$\boldsymbol{A}$$ is equivalent to conjugation by a permutation matrix $$\boldsymbol{P}$$. In other words, the reordered matrix can be written as $$\boldsymbol{A}_r = \boldsymbol{P}\boldsymbol{A}\boldsymbol{P}^*$$ and $$\boldsymbol{A}_r^{-1} = \boldsymbol{P}\boldsymbol{A}^{-1}\boldsymbol{P}^*$$.

Since the spectral norm is unitarily invariant and permutation matrices are unitary, the $$\ell^2$$ condition number $$\kappa(\boldsymbol{A}_r) = \|\boldsymbol{A}_r\|_2\|\boldsymbol{A}_r^{-1}\|_2 = \|\boldsymbol{P}\boldsymbol{A}\boldsymbol{P}^*\|_2\|\boldsymbol{P}\boldsymbol{A}^{-1}\boldsymbol{P}^*\|_2 = \|\boldsymbol{A}\|_2\|\boldsymbol{A}^{-1}\|_2 = \kappa(\boldsymbol{A}).$$

• Not only does the condition number not change under permutation, but the eigenvalues themselves also do not change. – Wolfgang Bangerth Aug 4 '19 at 20:34