Relative perturbation theory: IV. sin 2θ theorems Original Research Article

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+ΔA. 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 image and the new one image but instead bound the difference between image and its reflection image where the mirror is image and J reverses image , the orthogonal complement of image . The double angle bounds are proportional to the departure from the identity and from orthogonality of the matrix image . Note that image is invariant under the transformation D→D/α for α≠0, whereas the single angle theorems give bounds proportional to Dʹs departure from the identity and from orthogonality. The corresponding results for the singular value problem when a (nonsquare) matrix B is perturbed to image are also presented.