An analysis of the Rayleigh–Ritz method for approximating eigenspaces
Zhongxiao Jia, G. W. Stewart · Mathematics of Computation · 2000
This paper concerns the Rayleigh–Ritz method for computing an approximation to an eigenspace X \mathcal {X} of a general matrix A A from a subspace W \mathcal {W} that contains an approximation to X \mathcal {X} . The method produces a pair ( N , X ~ ) (N, \tilde X) that purports to approximate a pair ( L , X ) (L, X) , where X X is a basis for X \mathcal {X} and A X = X L AX = XL . In this paper we consider the convergence of ( N , X ~ ) (N, \tilde X) as the sine ϵ \epsilon of the angle between X \mathcal {X} and W \mathcal {W} approaches zero. It is shown that under a natural hypothesis — called the uniform separation condition — the Ritz pairs