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


1 Answer 1


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 $$\kappa(\boldsymbol{A}_r) = \|\boldsymbol{A}_r\|_2\|\boldsymbol{A}_r^{-1}\|_2 = \|\boldsymbol{P}\boldsymbol{A}\boldsymbol{P}^*\|_2\|\boldsymbol{P}\boldsymbol{A}^{-1}\boldsymbol{P}^*\|_2 = \|\boldsymbol{A}\|_2\|\boldsymbol{A}^{-1}\|_2 = \kappa(\boldsymbol{A}).$$

  • 5
    $\begingroup$ Not only does the condition number not change under permutation, but the eigenvalues themselves also do not change. $\endgroup$ Commented Aug 4, 2019 at 20:34

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