J Austral Math Soc Ser B 36 pp381--388, 1995.
(Received 3 July 1993; revised 16 September 1993)
The heart of the Lanczos algorithm is the systematic generation of
orthonormal bases of invariant subspaces of a perturbed matrix. The
perturbations involved are special since they are always rank-1 and are the
smallest possible in certain senses. These minimal perturbation properties are
extended here to more general cases.
Rank-1 perturbations are
also shown to be closely connected to inverse iteration, and thus provide a
novel explanation of the global convergence phenomenon of Rayleigh quotient
iteration.
Finally, we show that the restriction to a Krylov
subspace of a matrix differs from the restriction of its inverse by a rank-1
matrix.