using LinearAlgebra, PyPlot

t = 1
[cosh(t) sinh(t)
 sinh(t) cosh(t)]

exp([0 1; 1 0]*t)

t = 20
[cosh(t) sinh(t); sinh(t) cosh(t)]

exp([0 1; 1 0]*20)

exp(20)/2 * [1 1; 1 1]

A = randn(5,5)

exp(A)

series = I + A # first two terms
term = A
for n = 2:100
    term = term*A / n # compute Aⁿ / n! from the previous term Aⁿ⁻¹/(n-1)!
    series = series + term
end
series

λ, X = eigen(A)
X * Diagonal(exp.(λ)) / X

real(X * Diagonal(exp.(λ)) / X) # get rid of tiny imaginary parts

B = randn(5,5)
exp(A) * exp(B)

exp(A + B)

exp(A) * exp(2A)

exp(3A)

exp(2A) * exp(A)

inv(exp(A))

exp(-A)

C = [ 0     0    1 0
      0     0    0 1
     -0.02  0.01 0 0
      0.01 -0.02 0 0 ]
Δt = 1.0
T = exp(C*Δt)  # propagator matrix

x₀ = [0.0,0,1,0]  # initial condition

# loop over 300 timesteps and keep track of x₁(t)
x = x₀
x₁ = [ x₀[1] ]
for i = 1:300
    x = T*x     # repeatedly multiply by T
    push!(x₁, x[1])  # & store current x₁(t) in the array x₁
end

plot((0:300)*Δt, x₁, "r.-")
xlabel("time \$t\$")
ylabel("solution \$x_1(t)\$")
grid()

eigvals(exp(A*Δt))

λ = eigvals(A)
exp.(λ * Δt)