using LinearAlgebra
A=rand(5,5) .- 0.5
println("The random matrix is")
display(A)

x0=rand(5)
x=x0
for n=1:10000
    global x
    y=A*x
    x=y/norm(y)
end
println("\nAn approximate eigenvector is")
display(x)

e1=[1,0,0,0,0]
v=(x-e1)/norm(x-e1)
H=I - 2*v*v'
println("\nThe Hausholder reflector is")
display(H)

HAH=H*A*H
println("\nThe block matrix is")
display(HAH)

B=HAH[2:5,2:5]
println("\nThe deflated matrix B is")
display(B)

println("\nChecking...")
println("\nThe eigenvalues of A are")
display(eigvals(A))
println("\nThe eigenvalues of B are")
display(eigvals(B))

AA=copy(A)
for k=1:10000
    global AA
    z=qr(AA)
    AA=z.R*z.Q
end
println("\nTry the QR method on same matrix")
display(AA)