Relative Perturbation Theory : IV

The double angle theorems of Davis and Kahan bound the change in an invariant subspace when a Hermitian matrix A is subject to an additive perturbation A ! e A = A+A. This paper supplies analogous results when A is subject to a congruential, or multiplicative, perturbation A ! e A = D AD. The relative gaps that appear in the bounds involve the spectrum of only one matrix, either A or e A, in contrast to the gaps that appear in the single angle bounds. The double angle theorems do not directly bound the diierence between the old invariant subspace S and the new one e S but instead bound the diierence between e S and its reeection J e S where the mirror is S and J reverses S ? , the orthogonal complement of S. The double angle bounds are proportional to the departure from the identity and from orthogonality of the matrix e D def = D ?1 JDJ. Note that e D is invariant under the transformation D ! D== for 6 = 0, whereas the single angle theorems give bounds proportional to D's departure from the identity and from orthogonality. The corresponding results for the singular value problem when a (nonsquare) matrix B is perturbed to e B = D 1 BD 2 are also presented. Abstract The double angle theorems of Davis and Kahan bound the change in an invariant subspace when a Hermitian matrix A is subject to an additive perturbation A ! e A = A + A. This paper supplies analogous results when A is subject to a congruential, or multiplicative, perturbation A ! e A = D AD. The relative gaps that appear in the bounds involve the spectrum of only one matrix, either A or e A, in contrast to the gaps that appear in the single angle bounds. The double angle theorems do not directly bound the diierence between the old invariant subspace S and the new one e S but instead bound the diierence between e S and its reeection J e S where the mirror is S and J reverses S ? , the orthogonal complement of S. The double angle bounds are proportional to the departure from the identity and from orthogonality of the matrix e D def = D ?1 JDJ. Note that e D is invariant under the transformation D ! D== …