Linear Algebra ® Julia

# 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))