Get an eigvec by eigval, using eigvecs, only allow SymTridiagonal?

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The eigvecs(A, λ) method is currently only implemented for SymTridiagonal matrices. For other matrix types there is currently only eigvecs(A).

In principle, we could easily extend this to real-symmetric/Hermitian matrices, since they can be transformed into real SymTridiagonal matrices by the Hessenberg factorization. This should work:

using LinearAlgebra
import LinearAlgebra: eigvecs, RealHermSymComplexHerm

function eigvecs(A::RealHermSymComplexHerm, λ::AbstractVector{<:Real})
    F = hessenberg(A) # transform to SymTridiagonal form
    X = eigvecs(F.H, λ)
    return F.Q * X    # transform eigvecs of F.H back to eigvecs of A
end

For only computing a small subset of the eigenvectors, it seems to be a couple times faster than eigvecs, especially if you don’t count the cost of the eigenvalues (e.g. you already have them for some other reason).

Might be worth putting together a PR to LinearAlgebra.jl if you are interested in this functionality? eigvecs(A::Hermitian, eigvals) method? · Issue #1248 · JuliaLang/LinearAlgebra.jl · GitHub

No. nullspace employs an SVD, which is as costly as computing all the eigenvectors and eigenvalues with eigen. (In fact, the SVD calls eigen for Hermitian matrices.)

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