Absolute and relative perturbation bounds are derived for angles between invariant subspaces of complex square matrices, in the two-norm and in the Frobenius norm. The absolute bounds can be considered extensions of Davis and Kahan's sinθ theorem to general matrices and invariant subspaces of any dimension. The relative bounds are the most general relative bounds for invariant subspaces because they place no restrictions on the matrix or the perturbation. When the perturbed subspace has dimension one, the relative bound is implied by the absolute bound. © 2000 Elsevier Science Inc.
Ipsen, I. C. F. (2000). Absolute and relative perturbation bounds for invariant subspaces of matrices. Linear Algebra and Its Applications, 309(1–3), 45–56. https://doi.org/10.1016/S0024-3795(99)00104-4