Relative perturbation theory: IV. sin 2θ theoremsø

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 → Ã = A+1A. This paper supplies analogous results when A is subject to a congruential, or multiplicative, perturbation A → Ã = D∗AD. The relative gaps that appear in the bounds involve the spectrum of only one matrix, either A or Ã, in contrast to the gaps that appear in the single angle bounds. The double angle theorems do not directly bound the difference between the old invariant subspace S and the new one S̃ but instead bound the difference between S̃ and its reflection JS̃ 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 D̃ def = D−1JDJ . Note that D̃ is invariant under the transformation D → D/α for α / = 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 B̃ = D∗ 1BD2 are also presented. © 2000 Elsevier Science Inc. All rights reserved.