using LinearAlgebra

A = [0.9 0.2
     0.1 0.8]

eigvals(A)

A^100 * [17, 4]

cx₀ = A^1000 * [17, 4]

A * cx₀

o = [1,1]
o'A

A' * o

A - 1*I

(A - I) * [2,1]

# The following code prints a sequence of Aⁿx values
# for n=0,1,2,… nicely formatted with LaTeX.

x = [3, 0]
pmatrix(x) = string("\\begin{pmatrix} ", round(x[1],digits=3), "\\\\", round(x[2],digits=3), "\\end{pmatrix}")
buf = IOBuffer()
println(buf, "\$")
for k = 1:6
    print(buf, pmatrix(x), " \\longrightarrow ")
    for i = 1:4
        x = A*x
        print(buf, pmatrix(x), " \\longrightarrow ")
    end
    println(buf, "\\\\")
end
println(buf, "\$")
display("text/latex", String(take!(buf)))

A^100

P = [0 1
     1 0]

[P^n * [1,0] for n = 0:5]

eigvals(P)

X = eigvecs(P) # the eigenvectors

X ./ X[1,:]' # normalize the first row to be 1, to resemble our hand solutions

M = rand(5,5) # random entries in [0,1]

sum(M,dims=1) # not Markov yet

M = M ./ sum(M,dims=1)

sum(M,dims=1)

eigvals(M)

abs.(eigvals(M))

x = rand(5)
x = x / sum(x) # normalize x to have sum = 1

M^100 * x

sum(x), sum(M^100 * x) # still = 1

λ, X = eigen(M)
X[:,end] / sum(X[:,end]) # eigenvector for λ=1, normalized to sum=1