The tan θ theorem with relaxed conditions

The Davis-Kahan tanθ theorem bounds the tangent of the angles between an approximate and an exact invariant subspace of a Hermitian matrix. When applicable, it gives a sharper bound than the sinθ theorem. However, the tanθ theorem requires more restrictive conditions on the spectrums, demanding that the entire approximate eigenvalues (Ritz values) lie above (or below) the set of exact eigenvalues corresponding to the orthogonal complement of the invariant subspace. In this paper we show that the conditions of the tanθ theorem can be relaxed, in that the same bound holds even when the Ritz values lie both below and above the exact eigenvalues, but not vice versa.