  Convergence analysis of block iterative solvers for Hermitian eigenvalue
problems and the closely related research on properties of matrix-based signal
filters are challenging, and attract increasing attention due to their recent
applications in spectral data clustering and graph-based signal processing. We
combine majorization-based techniques pioneered for investigating the
Rayleigh-Ritz method in [SIAM J. Matrix Anal. Appl., 31 (2010), pp. 1521-1537]
with tools of classical analysis of the block power method by Rutishauser
[Numer. Math., 13 (1969), pp. 4-13] to derive convergence rate bounds of an
abstract block iteration, wherein tuples of tangents of principal angles or
relative errors of Ritz values are bounded using majorization in terms of
arranged partial sums and tuples of convergence factors. Our novel bounds are
robust in presence of clusters of eigenvalues, improve some previous results,
and are applicable to most known block iterative solvers and matrix-based
filters, e.g., to block power, Chebyshev, and Lanczos methods combined with
shift-and-invert approaches and polynomial filtering.
