An overview of relative sin Θ theorems for invariant subspaces of complex matrices

Relative perturbation bounds for invariant subspaces of complex matrices are reviewed, with emphasis on bounding the sines of the largest principal angle between two subspaces, i.e. sin Θ theorems. The goal is to provide intuition, as well as an idea for why the bounds hold and why they look the way they do. Relative bounds have the advantage of being better at exploiting structure in a perturbation than absolute bounds. Therefore the reaction of subspaces to relative perturbations can be different than to absolute perturbations. In particular, there are certain classes of relative perturbations to which subspaces of indefinite Hermitian matrices can be more sensitive than subspaces of definite matrices.