The concept of principal angles or canonical angles, between subspaces, is important theoretically and has many practical applications. In functional analysis the largest principal angle or gap is used to bound the perturbation of a closed linear operator. There are applications relevant to computation of canonical correlations in statistics, applications in information retrieval, in computation of eigenspaces, especially involving the analysis of the influence of changes in the trial subspace in the Rayleigh--Ritz method, and in Grassmannanian spaces where principal angles are used to define the chordal distance between subspaces.

This talk presents several new perturbation theorems involving a general inequality and majorization of principal angles for subspaces of a finite dimensional space that holds for arbitrary unitarily invariant norms. One new theorem involves perturbations where the absolute value of the difference of the squares of the cosines/sines are majorized by the sines of the angles between the perturbed subspaces.

The other theorems involve upper bounds for proximity of the Ritz values in terms of the proximity of the trial subspaces without making an assumption that the trial subspace is close to an invariant subspace. The main result is that the absolute value of the perturbations in the Ritz values is bounded by a constant times the gap between the original trial subspace and its perturbation. The constant is the spread in the matrix spectrum, i.e. the difference between the largest and the smallest eigenvalues of the matrix. We then generalize this result to arbitrary unitarily invariant norms, but we have to increase the constant by a factor of square root of two.