Interactive eigenvectors¶

function plot_image(M, x, tol=0.05)
    
    new_x = M * x
    
    unit_circle = [(cos(θ), sin(θ)) for θ in 0:0.01:2π]
    ellipse = [M*[cos(θ), sin(θ)] for θ in 0:0.01:2π];

    
    plot([(0, 0), (x[1], x[2])], aspect_ratio=1, lw=2, leg=false, c=:green, arrow=0.4)
    plot!([(0, 0), (new_x[1], new_x[2])], lw=3, c=:blue, arrow=0.4)

    plot!(unit_circle, ls=:dash, c=:green)
    plot!(first.(ellipse), last.(ellipse), ls=:dash, c=:blue)


    # highlight when an eigenvector by changing color:
    
    ang1 = atan(x[2], x[1])
    ang2 = atan(new_x[2], new_x[1])
    
    if abs(ang1 - ang2) < tol || abs(ang1 - ang2 + π) < tol || abs(ang1 - ang2 - π) < tol
        plot!([(0, 0), (new_x[1], new_x[2])], lw=4, c=:magenta)
        plot!([(0, 0), (x[1], x[2])], lw=4, c=:magenta)
    end
    
    scatter!([0], [0])

    xlims!(-2, 2)
    ylims!(-2, 2)
end

a = slider(-2:0.1:2, label="a")
b = slider(-2:0.1:2, label="b")
c = slider(-2:0.1:2, label="c")
d = slider(-2:0.1:2, label="d")


sliders = vbox( hbox(a, b),
                hbox(c, d)
            )

r = -2:0.01:2

θ_vals = range(-2π, 2π, length = 500)
formatted_vals = map(t -> "$(floor(t/π, digits=2))π", θ_vals)

#@manipulate for a in r, b in r, c in r, d in r, θ in 

θ = slider(θ_vals, formatted_vals, label="θ")

p = map(a, b, c, d, θ) do a, b, c, d, θ

    M = [a b; c d]
   
    x = [cos(θ), sin(θ)]
    
    plot_image(M, x)

end

vbox(sliders, θ, p)

@manipulate for σ₁ in 0:0.01:2, σ₂ in 0:0.01:2, θ in 0:0.01:2π, ϕ in 0:0.01:2π, angle=-2π:0.01:2π

    # when phi=theta have U = V so a symmetric matrix
    
    U = [cos(θ) sin(θ); -sin(θ) cos(θ)]
    V = [cos(ϕ) sin(ϕ); -sin(ϕ) cos(ϕ)]

    Σ = Diagonal([σ₁, σ₂])

    M = U * Σ * V'
   
    x = [cos(angle), sin(angle)]

    plot_image(M, x, 0.01)
    
    title!("evals = $(eigvals(M))")
end