# some packages we will use
using LinearAlgebra
using SparseArrays
using Images
using MAT
A = rand(10,10); # created a random matrix of size 10-by-10
Atranspose = A' # matrix transpose
A = A*Atranspose; # matrix multiplication
@show A[11] == A[1,2];
A[11] == A[1, 2] = true
b = rand(10); #created a random vector of size 10
x = A\b; #x is the solutions to the linear system Ax=b
@show norm(A*x-b);
norm(A * x - b) = 2.213691169774188e-13
@show typeof(A)
@show typeof(b)
@show typeof(rand(1,10))
@show typeof(Atranspose);
typeof(A) = Matrix{Float64}
typeof(b) = Vector{Float64}
typeof(rand(1, 10)) = Matrix{Float64}
typeof(Atranspose) = Adjoint{Float64, Matrix{Float64}}
Matrix{Float64} == Array{Float64,2}
Vector{Float64} == Array{Float64,1}
Atranspose
?adjoint
search: adjoint adjoint! Adjoint
Atranspose.parent
sizeof(A)
# To actually copy the matrix:
B = copy(Atranspose)
sizeof(B)
?\
search: \
luA = lu(A)
norm(luA.L*luA.U - luA.P*A)
qrA = qr(A)
norm(qrA.Q*qrA.R - A)
isposdef(A)
cholA = cholesky(A)
norm(cholA.L*cholA.U - A)
cholA.L
cholA.U
factorize(A)
?factorize
search: factorize Factorization factorial
?diagm
search: diagm spdiagm diag diagind Diagonal isdiag Bidiagonal Tridiagonal
# convert(Diagonal{Int64,Array{Int64,1}},diagm([1,2,3]))
Diagonal([1,2,3])
I(3)
using SparseArrays
S = sprand(5,5,2/5)
S.rowval
Matrix(S)
S.colptr
S.m
using Pkg;
Pkg.add("ImageMagick")
Updating registry at `~/.julia/registries/General`
Resolving package versions...
Installed MKL_jll ───────── v2021.1.1+2
Installed Libtiff_jll ───── v4.3.0+0
Installed ImageMagick ───── v1.2.2
Installed FFTW_jll ──────── v3.3.9+8
Installed ImageMagick_jll ─ v6.9.10-12+3
Installed JpegTurbo_jll ─── v2.1.0+0
Installed libpng_jll ────── v1.6.38+0
Updating `~/.julia/environments/v1.6/Project.toml`
[6218d12a] + ImageMagick v1.2.2
Updating `~/.julia/environments/v1.6/Manifest.toml`
[6218d12a] + ImageMagick v1.2.2
[f5851436] ↑ FFTW_jll v3.3.9+7 ⇒ v3.3.9+8
[c73af94c] + ImageMagick_jll v6.9.10-12+3
[aacddb02] + JpegTurbo_jll v2.1.0+0
[89763e89] + Libtiff_jll v4.3.0+0
[856f044c] ↑ MKL_jll v2021.1.1+1 ⇒ v2021.1.1+2
[b53b4c65] + libpng_jll v1.6.38+0
Precompiling project...
✓ libpng_jll
✓ JpegTurbo_jll
✓ FFTW_jll
✓ MKL_jll
✓ Libtiff_jll
? FFTW
✓ ImageMagick_jll
✓ ImageMagick
7 dependencies successfully precompiled in 14 seconds (90 already precompiled)
1 dependency failed but may be precompilable after restarting julia
X1 = load("/work/data/khiam-small.jpg")
@show typeof(X1)
X1[1,1] # this is pixel [1,1]
typeof(X1) = Matrix{RGB{N0f8}}
Xgray = Gray.(X1)
R = map(i->X1[i].r,1:length(X1))
R = Float64.(reshape(R,size(X1)...))
G = map(i->X1[i].g,1:length(X1))
G = Float64.(reshape(G,size(X1)...))
B = map(i->X1[i].b,1:length(X1))
B = Float64.(reshape(B,size(X1)...))
;
Z = zeros(size(R)...) # just a matrix of all zeros of equal size as the image
RGB.(Z,G,Z)
Xgrayvalues = Float64.(Xgray)
SVD_V = svd(Xgrayvalues)
norm(SVD_V.U*diagm(SVD_V.S)*SVD_V.V' - Xgrayvalues)
# use the top 4 singular vectors/values to form a new image
u1 = SVD_V.U[:,1]
v1 = SVD_V.V[:,1]
img1 = SVD_V.S[1]*u1*v1'
i = 2
u1 = SVD_V.U[:,i]
v1 = SVD_V.V[:,i]
img1 += SVD_V.S[i]*u1*v1'
i = 3
u1 = SVD_V.U[:,i]
v1 = SVD_V.V[:,i]
img1 += SVD_V.S[i]*u1*v1'
i = 4
u1 = SVD_V.U[:,i]
v1 = SVD_V.V[:,i]
img1 += SVD_V.S[i]*u1*v1'
Gray.(img1)
i = 1:100
u1 = SVD_V.U[:,i]
v1 = SVD_V.V[:,i]
img1 = u1*spdiagm(0=>SVD_V.S[i])*v1'
Gray.(img1)
norm(Xgrayvalues-img1)
M = matread("/work/data/face_recog_qr.mat")
q = reshape(M["V2"][:,1],192,168)
Gray.(q)
b = q[:]
A = M["V2"][:,2:end]
x = A\b #Ax=b
Gray.(reshape(A*x,192,168))
norm(A*x-b)
qv = q+rand(size(q,1),size(q,2))*0.5
qv = qv./maximum(qv)
Gray.(qv)
b = qv[:];
x = A\b
norm(A*x-b)
Gray.(reshape(A*x,192,168))