✔ Hermitian tridiagonal arrays · helpdesk (published)

real_form(A::Hermitian{T, <:Tridiagonal}) where {T <: Real} = (;T = A,  S = Diagonal(ones(T, size(A, 1))), )
function real_form(A::Hermitian{T, <:Tridiagonal}) where {T}
    (; dl, d) = parent(A)
    N = length(d)
    if N == 1
        return (; T = SymTridiagonal(real.(d), real.(dl)), S = Diagonal([one(T)]))
    end
    S = Vector{ComplexF64}(undef, N)
    E = dl
    Er = abs.(E)
    S[1] = 1
    S[2] = Er[1] == 0 ? one(T) : S[1] * E[1]/Er[1]
    for i ∈ 2:N-1
        S[i+1] = Er[i] == 0 ? one(T) : S[i] * E[i]/Er[i]
    end
    (; T = SymTridiagonal(real.(d), Er), S = Diagonal(S))
end

This will produce a real SymTridiagonal matrix T which has the same eigenvalues as the hermitian matrix A, as well as a diagonal matrix S such that S * T * S' ≈ A, so that if U are the eigenvectors of T, then

S * U * Diagonal(eigvals(T)) * U' * S' ≈ A

Stream: helpdesk (published)

Topic: ✔ Hermitian tridiagonal arrays

Mason Protter (Apr 26 2023 at 22:37):

Notification Bot (Apr 26 2023 at 22:38):