using Plots

# nice viz for matrices
function lookat(A; redrow=0, rounding=2, showtext=true)
  n = size(A,1)
  plot(legend=false, axis=false)
  rowcolor = redrow > 0 ? :red : :black  
  for i=1:n, j=1:n  
      scatter!( [j],[i], ann=  showtext ? (j,i,round(A[i,j],digits=rounding), :white ) : (j,i,"") ,
                color=abs(A[i,j]) > .0001 ? (i==redrow ? rowcolor : :black) : :white, 
                marker=:square, markersize=30, aspectratio=1, yflip=true, yaxis=[.5,n+.5],xaxis=[.5,n+.5])
  end
  plot!()
end

A = rand(1.0:9,4,4)

L = fill(0.0,4,4)

lookat(A)

L[2,1] = A[2,1] / A[1,1]

# A[2,:] = A[2,:] - L[2,1] * A[1,:]
A[2,:] -= L[2,1] * A[1,:] # subtract that multiple of the first row from the second
lookat(A, redrow=2)

L[3,1] = A[3,1] / A[1,1]

A[3,:] -= L[3,1] * A[1,:]
lookat(A, redrow=3)

L[4,1] = A[4,1] / A[1,1]
A[4,:] -= L[4,1] * A[1,:]
lookat(A, redrow=4)

L[3,2] = A[3,2] / A[2,2]

A[3,:] -= L[3,2] * A[2,:]; lookat(A, redrow=3)

L[4,2] = A[4,2] / A[2,2]; A[4,:] -= L[4,2] * A[2,:]; lookat(A, redrow=4)

n = 5
A = randn(n,n)
L = fill(0.0,n,n)
display(lookat(A))
for j=1:n, i=(j+1):n
        L[i,j] = A[i,j]/A[j,j]
        A[i,:] -= L[i,j] * A[j,:]
        display(lookat(A,redrow=i))
end

n = 5
A = randn(n,n)
L = fill(0.0,n,n)
Akeep = [copy(A)]
row = [0]
display(lookat(A))
for j=1:n, i=(j+1):n
        L[i,j] = A[i,j]/A[j,j]
        A[i,:] -= L[i,j] * A[j,:]
        push!(Akeep,copy(A))
        push!(row,i)
end

using Interact

@manipulate for i=slider(1:length(Akeep),value=1)
      lookat(Akeep[i],redrow=row[i],showtext=true)
end

using LinearAlgebra

U =  Akeep[end]
A =  Akeep[1]
L += I

L*U

A

L*U ≈ A