Λ = diagm([0.1, 0.2, 0.4, 0.5])

Λ * [10
     0
     0
     0]

Λ * [0
     10
     0
     0]

x = [1,1,1,1]
Λ * x

Λ^10 * x

Λ^100

using Interact
@manipulate for n = slider(0:100, value=0)
    y = Λ^n * x
    y / norm(y)
end

inv(Λ)

X = [ 1 0 1  0
      2 1 0  1
      0 1 1  0
      1 0 0 -1]
rank(X)

A = X * Λ / X

A * X[:, 1]

A * X[:, 2]

x

c = X\x

A^100 * x

(A^100*x) / norm(A^100*x)

A^100

inv(A)

X * inv(Λ) * inv(X)

eigenvalues, eigenvectors = eig(A)

eigenvalues

eigenvectors

round.(X \ eigenvectors, 5) # X⁻¹ * eigenvectors, rounded to 5 digits

using PyPlot
λ = linspace(0,0.6,100)
plot(λ, [det(A - λ*I) for λ in λ], "r-")
plot(λ, 0*λ, "k--")
plot(diag(Λ),diag(Λ)*0, "bo")
xlabel(L"\lambda")
ylabel(L"\det(A - \lambda I)")
title("characteristic polynomial")