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- W2000614097 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→ 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 * 1 BD 2 are also presented." @default.
- W2000614097 created "2016-06-24" @default.
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- W2000614097 date "2000-05-01" @default.
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- W2000614097 title "Relative perturbation theory: IV. sin 2θ theorems" @default.
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- W2000614097 doi "https://doi.org/10.1016/s0024-3795(00)00077-x" @default.
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