
# https://gist.github.com/shashi/e1ade8f935802d83ebc2 # inspiration
using LinearAlgebra: eigen, Diagonal, opnorm
using Interact # @manipulate
using Plots; default(markerstrokecolor=:auto)


A = [2 1; 1 3] # symmetric matrix: principal axes are eigenvectors
#A = [1 -1; 1 -1] # special outer product, cf HW01!
#A = [1 1; 1 1] # symmetric outer product
#A = [2 1; 0 2] # asymmetric matrix, not normal
#A = [cos(π/3) sin(π/3); -sin(π/3) cos(π/3)] # rotation, normal, complex eigvecs!
#A = [1 1; 0 1] # "defective" matrix: does not have lin. ind. V
display(isapprox(A'A, A*A') ? "normal!" : "not normal")
A

"normal!"

2×2 Matrix{Int64}:
 2  1
 1  3


Φ = range(0, 2π, 181)
xc = [cos.(Φ) sin.(Φ)]'
yc = A * xc

@manipulate for θ in slider(Φ, value=Φ[1])
#anim = @animate for θ in Φ
#for θ in (π/2,)
    x = [cos(θ), sin(θ)] # vector on unit circle
    y = A * x # matrix-vector multiplication of interest
    plot(xc[1,:], xc[2,:], line=(:blue,:dash), label="",
        xlim = (-1.1, 1.1) .* opnorm(A), xlabel="x1",
        ylim = (-1.1, 1.1) .* opnorm(A), ylabel="x2",
        aspect_ratio = :equal)
    plot!(yc[1,:], yc[2,:], line=(:purple,:dash), label="")
    plot!([0, x[1]], [0, x[2]], color=:blue, label="")
    plot!([0, y[1]], [0, y[2]], color=:purple, label="")
    scatter!([0, x[1]], [0, x[2]], color=:blue, label="x")
    scatter!([0, y[1]], [0, y[2]], color=:purple, label="y")
end


plot!() # for saving as html


#savefig("tmp.pdf") # save figure to pdf file


#gif(anim, "03_eigshow1_2113.gif")
#gif(anim, "03_eigshow1_2102.gif")
#gif(anim, "03_eigshow1_1101.gif")


# http://docs.juliaplots.org/latest/animations/
# example fails 2018-08-20
if false
    using Iterators
    itr = repeatedly(()->rand(10), 20)
    animate(itr, ylims=(0,1), c=:red, fps=5)
end


# saves a file - doesn't display it
if false
    @gif for i=1:10
        plot((1:9).^i)
    end
end


(d,V) = eigen(A)
display(V)
display(V'V)
display(V * Diagonal(d) * V')
@show d;

2×2 Matrix{Float64}:
 -0.850651  0.525731
  0.525731  0.850651

2×2 Matrix{Float64}:
 1.0  0.0
 0.0  1.0

2×2 Matrix{Float64}:
 2.0  1.0
 1.0  3.0

d = [1.381966011250105, 3.618033988749895]


ϕ=π/4; (pi, π, cos(ϕ))

(π, π, 0.7071067811865476)

Eigenvector illustration for 2×2 matrix¶