Relative perturbation theory: IV. sin 2 [formula omitted] theorems

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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→ A ̃ =A+ΔA . This paper supplies analogous results when A is subject to a congruential, or multiplicative, perturbation A→ A ̃ =D *AD . The relative gaps that appear in the bounds involve the spectrum of only one matrix, either A or A ̃ , 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 S ̃ but instead bound the difference between S ̃ and its reflection J 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 D ̃ = def D −1 JDJ . Note that D ̃ 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 B ̃ =D * 1BD 2 are also presented.