384 lines
8.0 KiB
Plaintext
Generated
384 lines
8.0 KiB
Plaintext
Generated
{
|
||
"cells": [
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 1,
|
||
"id": "a88398a0-03e1-458f-ba48-518fb151b1b3",
|
||
"metadata": {},
|
||
"outputs": [],
|
||
"source": [
|
||
"using Plots\n",
|
||
"using LinearAlgebra"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 2,
|
||
"id": "deac13e6-b3dd-4c8d-9c15-504052e60c6d",
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"3×3 Matrix{Float64}:\n",
|
||
" 0.693988 0.0373642 0.306196\n",
|
||
" 0.475249 0.0523086 0.525352\n",
|
||
" 0.275428 0.0212671 0.431897"
|
||
]
|
||
},
|
||
"execution_count": 2,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"A = rand(3,3)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 3,
|
||
"id": "a7cfa41c-b406-475b-8a8e-bc56cab68cfa",
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"SVD{Float64, Float64, Matrix{Float64}, Vector{Float64}}\n",
|
||
"U factor:\n",
|
||
"3×3 Matrix{Float64}:\n",
|
||
" -0.649854 0.735055 0.193349\n",
|
||
" -0.624454 -0.371326 -0.687149\n",
|
||
" -0.433297 -0.567284 0.700316\n",
|
||
"singular values:\n",
|
||
"3-element Vector{Float64}:\n",
|
||
" 1.1253009779161582\n",
|
||
" 0.27878010490969024\n",
|
||
" 0.013851237498536341\n",
|
||
"Vt factor:\n",
|
||
"3×3 Matrix{Float64}:\n",
|
||
" -0.770554 -0.0587937 -0.634658\n",
|
||
" 0.636347 -0.0144316 -0.771268\n",
|
||
" 0.0361865 -0.998166 0.0485336"
|
||
]
|
||
},
|
||
"execution_count": 3,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"U, S, V = svd(A) #singular decomposition"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 4,
|
||
"id": "c6fdc076-bfc8-46e9-97c2-5427f4c49395",
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"3×3 BitMatrix:\n",
|
||
" 1 0 0\n",
|
||
" 0 1 0\n",
|
||
" 0 0 1"
|
||
]
|
||
},
|
||
"metadata": {},
|
||
"output_type": "display_data"
|
||
},
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"3×3 BitMatrix:\n",
|
||
" 1 0 0\n",
|
||
" 0 1 0\n",
|
||
" 0 0 1"
|
||
]
|
||
},
|
||
"metadata": {},
|
||
"output_type": "display_data"
|
||
},
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"3×3 BitMatrix:\n",
|
||
" 0 0 0\n",
|
||
" 0 0 0\n",
|
||
" 0 0 0"
|
||
]
|
||
},
|
||
"metadata": {},
|
||
"output_type": "display_data"
|
||
}
|
||
],
|
||
"source": [
|
||
"display(U'*U .> 0.00001)\n",
|
||
"display(V'*V .> 0.00001)\n",
|
||
"display(A - (U * Diagonal(S) * V') .> 0.00001)"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 5,
|
||
"id": "37b7a415-d7ea-4d00-b704-7d79e562d899",
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"3-element Vector{Float64}:\n",
|
||
" 0.019506810588402184\n",
|
||
" 0.2433655332706062\n",
|
||
" 0.9153216003851385"
|
||
]
|
||
},
|
||
"metadata": {},
|
||
"output_type": "display_data"
|
||
},
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"3-element Vector{Float64}:\n",
|
||
" 1.1253009779161582\n",
|
||
" 0.27878010490969024\n",
|
||
" 0.013851237498536341"
|
||
]
|
||
},
|
||
"metadata": {},
|
||
"output_type": "display_data"
|
||
}
|
||
],
|
||
"source": [
|
||
"display(eigvals(A))\n",
|
||
"display(svd(A).S)\n",
|
||
"# singular values are more spread out"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 39,
|
||
"id": "5a4893c6-074c-4479-ac8a-9ad8db295d0a",
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"((3, 3), (3,), (5, 3))"
|
||
]
|
||
},
|
||
"metadata": {},
|
||
"output_type": "display_data"
|
||
},
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"((5, 3), (3,), (3, 3))"
|
||
]
|
||
},
|
||
"metadata": {},
|
||
"output_type": "display_data"
|
||
}
|
||
],
|
||
"source": [
|
||
"# svd exists also for rectangular matrices\n",
|
||
"U, S, V = svd(rand(3, 5))\n",
|
||
"display((size(U), size(S), size(V)))\n",
|
||
"U, S, V = svd(rand(5, 3))\n",
|
||
"display((size(U), size(S), size(V)))"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 9,
|
||
"id": "380565b8-f15d-4f3f-9a0f-85fc82809bdd",
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"3-element Vector{Float64}:\n",
|
||
" 2.302775637731995\n",
|
||
" 1.3027756377319948\n",
|
||
" 0.0"
|
||
]
|
||
},
|
||
"execution_count": 9,
|
||
"metadata": {},
|
||
"output_type": "execute_result"
|
||
}
|
||
],
|
||
"source": [
|
||
"svdvals([2 0 1;0 1 1; 0 0 0; 0 0 0]) # rank is 2 -> so 2 non zero values"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 23,
|
||
"id": "deb05175-6a64-4c40-ab5c-e614ef85797f",
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"Eigen{Float64, Float64, Matrix{Float64}, Vector{Float64}}\n",
|
||
"values:\n",
|
||
"3-element Vector{Float64}:\n",
|
||
" 4.440892098500626e-15\n",
|
||
" 1.697224362268007\n",
|
||
" 5.302775637731994\n",
|
||
"vectors:\n",
|
||
"3×3 Matrix{Float64}:\n",
|
||
" -0.333333 0.444872 -0.831251\n",
|
||
" -0.666667 -0.734656 -0.125841\n",
|
||
" 0.666667 -0.51222 -0.541467"
|
||
]
|
||
},
|
||
"metadata": {},
|
||
"output_type": "display_data"
|
||
},
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"3×3 Matrix{Float64}:\n",
|
||
" 4.0 0.0 2.0\n",
|
||
" 0.0 1.0 1.0\n",
|
||
" 2.0 1.0 2.0"
|
||
]
|
||
},
|
||
"metadata": {},
|
||
"output_type": "display_data"
|
||
}
|
||
],
|
||
"source": [
|
||
"A = [2 0 1;0 1 1; 0 0 0];\n",
|
||
"display(eigen(A' * A))\n",
|
||
"U, S, V = svd(A);\n",
|
||
"display(V * Diagonal(S)' * Diagonal(S) * V')"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 32,
|
||
"id": "60516aff-16aa-41e6-a56d-6cad04547e55",
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"3×3 Matrix{Float64}:\n",
|
||
" 1.0 -1.0 0.0\n",
|
||
" 0.0 1.0 0.0\n",
|
||
" 0.0 0.0 0.5"
|
||
]
|
||
},
|
||
"metadata": {},
|
||
"output_type": "display_data"
|
||
},
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"3×3 Matrix{Float64}:\n",
|
||
" 1.0 -1.0 0.0\n",
|
||
" -1.11022e-16 1.0 0.0\n",
|
||
" 0.0 0.0 0.5"
|
||
]
|
||
},
|
||
"metadata": {},
|
||
"output_type": "display_data"
|
||
}
|
||
],
|
||
"source": [
|
||
"A = [1 1 0; 0 1 0; 0 0 2];\n",
|
||
"display(inv(A))\n",
|
||
"F = svd(A);\n",
|
||
"display(F.V * Diagonal(F.S .^ -1) * F.U') # we can get the inverse from the svd"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 46,
|
||
"id": "cd9e606f-957d-4803-9ab8-f7b23c455e13",
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"(2.0, false)"
|
||
]
|
||
},
|
||
"metadata": {},
|
||
"output_type": "display_data"
|
||
},
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"\"frobenius norm: 2.6457513110645907\""
|
||
]
|
||
},
|
||
"metadata": {},
|
||
"output_type": "display_data"
|
||
}
|
||
],
|
||
"source": [
|
||
"display((opnorm(A), opnorm(A) == norm(A))) # use opnorm for the matrix norm\n",
|
||
"display(\"frobenius norm: $(norm(A))\")"
|
||
]
|
||
},
|
||
{
|
||
"cell_type": "code",
|
||
"execution_count": 57,
|
||
"id": "9c469ac2-4d2e-4eba-9c6e-c63ec1fc06ae",
|
||
"metadata": {},
|
||
"outputs": [
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"2×2 Matrix{Float64}:\n",
|
||
" 1.27357 1.80721\n",
|
||
" 2.87898 4.08529"
|
||
]
|
||
},
|
||
"metadata": {},
|
||
"output_type": "display_data"
|
||
},
|
||
{
|
||
"data": {
|
||
"text/plain": [
|
||
"1"
|
||
]
|
||
},
|
||
"metadata": {},
|
||
"output_type": "display_data"
|
||
}
|
||
],
|
||
"source": [
|
||
"# Eckhart-Young theorem\n",
|
||
"# the min X with respect to opnorm(A-X) with rank less than or equal to k is:\n",
|
||
"A = [1 2;3 4];\n",
|
||
"F = svd(A);\n",
|
||
"X = F.U[:,1] .* F.S[1, 1] * F.V[:, 1]';\n",
|
||
"display(X)\n",
|
||
"display(rank(X))"
|
||
]
|
||
}
|
||
],
|
||
"metadata": {
|
||
"kernelspec": {
|
||
"display_name": "Julia 1.9.3",
|
||
"language": "julia",
|
||
"name": "julia-1.9"
|
||
},
|
||
"language_info": {
|
||
"file_extension": ".jl",
|
||
"mimetype": "application/julia",
|
||
"name": "julia",
|
||
"version": "1.9.3"
|
||
}
|
||
},
|
||
"nbformat": 4,
|
||
"nbformat_minor": 5
|
||
}
|