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Proof. Suppose that U1, . . . , Um are invariant subspaces of T ∈ L(V ). We wish to show that U1 + · · ·+ Um is also an invariant subspace. Consider any ~v ∈ U1 + · · ·+ Um. By definition, we can write ~v = ~u1 + · · ·+ ~um, for some ~u1 ∈ U1, . . . , ~um ∈ Um. Since U1, . . . , Um are invariant subspaces, by definition, we know T~u1 ∈ U1 ⊂ U1 + · · ·+ Um, ..., T~um ∈ Um ⊂ U1 + · · ·+ Um. Hence...

Large scale eigenvalue computation is about approximating certain invariant subspaces associated with the interested part of the spectrum, and the interested eigenvalues are then extracted from projecting the problem by approximate invariant subspaces into a much smaller eigenvalue problem. In the case of the linear response eigenvalue problem (aka the random phase eigenvalue problem), it is th...

The performance of Krylov subspace eigenvalue algorithms for large matrices can be measured by the angle between a desired invariant subspace and the Krylov subspace. We develop general bounds for this convergence that include the effects of polynomial restarting and impose no restrictions concerning the diagonalizability of the matrix or its degree of nonnormality. Associated with a desired se...