A = randn(5,5)
U,σ,V = svd(A)
display(U), display(σ), display(V);

σ
Σ = Diagonal(σ)

round.(U'U,2)

round.(V'V,2)

σ

Σ=Diagonal(σ)
U*Σ*V'

A

A = randn(5,3)
U,σ,V = svd(A,thin=false) # thin=false gives the basic svd, 
                          # often the default (thin=true) is more useful
display(U), display(σ), display(V);

Σ = zeros(A) # Make Σ the size of A
for i=1:length(σ) Σ[i,i]=σ[i] end # put the singular values on the diagonal

display(A)
U*Σ*V'

Σ

A = randn(3,5)
U,σ,V = svd(A,thin=false) # thin=false gives the basic svd, 
                          # often the default (thin=true) is more useful
display(U), display(σ), display(V)
Σ = zeros(A) # Make Σ the size of A
for i=1:length(σ) Σ[i,i]=σ[i] end # put the singular values on the diagonal
U*Σ*V'
A

A

Σ

A = randn(5,3)
U,σ,V = svd(A) # thin = true
display(U)
display(round(U'U,2))
display(A)
U*Diagonal(σ)*V'

A = randn(3,5)
U,σ,V = svd(A)
display(V)
display(round(V'V,2))
display(A)
U*Diagonal(σ)*V'

using Images, Interact
picture = download("http://web.mit.edu/jfrench/Public/gstrang.png")
pimage = load(picture)

p = Float64.(channelview(pimage)) # convert to an array

pr,pg,pb = p[1,:,:],p[2,:,:],p[3,:,:]
Ur,σr,Vr = svd(pr)
Ug,σg,Vg = svd(pg)
Ub,σb,Vb = svd(pb)

[colorview(RGB,pg,0*pg,0*pg);colorview(RGB,0*pg,pg,0*pg);colorview(RGB,0*pg,0*pg,pg)]

#Pkg.add("ImageMagick")

using ImageMagick

# Pkg.add("Plots")
# Pkg.build("PlotlyJS")

using Plots
plotly()

plot(σr,   color=:red,     yaxis=:log, legend=false)
plot!(σg,  color=:green, yaxis=:log)
plot!(σb,  color=:blue,  yaxis=:log)

using Interact

@manipulate for k=slider(1:40,value=1)
    p̂r = Ur[:,1:k]*Diagonal(σr[1:k])*Vr[:,1:k]'
    p̂g = Ug[:,1:k]*Diagonal(σg[1:k])*Vg[:,1:k]'
    p̂b = Ub[:,1:k]*Diagonal(σb[1:k])*Vb[:,1:k]'
    img = colorview(RGB, p̂r,p̂g,p̂b) 
end

pics = Dict{String,Array}()
pics["Ireland"] = ImageMagick.load(download("https://upload.wikimedia.org/wikipedia/commons/thumb/4/45/Flag_of_Ireland.svg/500px-Flag_of_Ireland.svg.png"))
pics["USA"] = ImageMagick.load(download("https://upload.wikimedia.org/wikipedia/en/thumb/a/a4/Flag_of_the_United_States.svg/640px-Flag_of_the_United_States.svg.png"))
pics["Klingon"] = ImageMagick.load(download("https://images-na.ssl-images-amazon.com/images/I/51ibu5dAb9L._SY550_.jpg"))


@manipulate for flag in collect(keys(pics)), k=slider(1:40,value=1)
    p = float.(channelview(pics[flag])) # convert to an array
    pr,pg,pb = p[1,:,:],p[2,:,:],p[3,:,:]
    Ur,σr,Vr = svd(pr)
    Ug,σg,Vg = svd(pg)
    Ub,σb,Vb = svd(pb)
    p̂r = Ur[:,1:k]*Diagonal(σr[1:k])*Vr[:,1:k]'
    p̂g = Ug[:,1:k]*Diagonal(σg[1:k])*Vg[:,1:k]'
    p̂b = Ub[:,1:k]*Diagonal(σb[1:k])*Vb[:,1:k]'
    colorview(RGB, p̂r,p̂g,p̂b) 
end

A = eye(5)
svdvals(A)

A = Diagonal(randn(5))

svdvals(A) # absolute value of diagonals
# svdvals always are in monotonically non-increasing order

Q,_ = qr(randn(5,5))

Q'Q

svdvals(Q)
# Q = Q * (Σ=I) * (V=I)'

D = Diagonal(randn(5))

svdvals(Q*D)

svdvals(D*Q)

# Notice if A = U Σ V' then
# A'A = U Σ V'V Σ' U'
# A'A = V Σ² V'
# A*A' = U Σ² U'

A = randn(5,3)

svdvals(A).^2 |> sort

eigvals(A'A) |> sort

eigvals(A*A') |> sort

U,σ,V = svd(A)

E,X = eig(A'A)
display(X)

V[:,5:-1:1]

E,X = eig(A*A')
display(X)

U[:,5:-1:1]

n(n-1)/2 + n + n(n-1)/2