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Let α be a complex eigenvector corresponding to (e^{ itheta}) and β be a complex eigenvector of the transposed matrix (H^{T}) corresponding to (e^{-itheta}), i.e., (Halpha= e^{ itheta_{0}}alpha), (H^{T}beta= e^{-itheta_{0}}beta).
Let q ∈ ≤ n be a complex eigenvector of A corresponding to the eigenvalue iω, i.e. Aq = iωq.
In the above section, we see that H has simple eigenvalues on the unit circle: λ 1, 2 = e ± i θ 0, θ 0 = arc cos 1 + ( p + q ) 2 − q 2 2 ( p + q ) 2. Let α be a complex eigenvector corresponding to e i θ and β be a complex eigenvector of the transposed matrix H T corresponding e − i θ, i.e. H α = e i θ 0 α, H T β = e − i θ 0 β.
Let q ∈ C n be a complex eigenvector corresponding to the eigenvalue λ 1, then we have A q = i α q, A q ¯ = − i α q ¯.
Let q ∈ C 3 ( k + 1 ) be a complex eigenvector corresponding to e i ω ∗ satisfying A q = e i ω ∗ q, A q ¯ = e − i ω ∗ q ¯.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com