### A Pluto.jl notebook ###
# v0.19.27

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
    quote
        local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
        local el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
        el
    end
end

# ╔═╡ 6bf0c7ba-1195-11ec-28bf-a5e13ac974c0
# packages
begin
using LinearAlgebra: svd, Diagonal, norm, opnorm, eigen
using PlutoUI: Slider, Select, CheckBox
using LaTeXStrings
using Plots: plot, plot!, default
default(markerstrokecolor=:auto, label="")
end

# ╔═╡ 6bf0c7ba-1195-11ec-3f14-1d11b490e8a2
md"## Illustration of SVD for 2×2 case  

2017-09-14, Jeff Fessler, University of Michigan

2020-08-06 Julia 1.5.0

2021-08-23 Julia 1.6.2

2021-09-09 Pluto"


# ╔═╡ e9ffa885-2885-4a9b-b82f-4ccf894e1095
# zoo of matrices as possible examples
zoo = [
[2 1; 0 3], # asymmetric matrix - no problem for SVD!
[1 1; 0 0], # rank1 matrix: u2 is not in range of A
[2 1; 0 2], # defective (not diagonalizable)
[cos(π/3) sin(π/3); -sin(π/3) cos(π/3)], # rotation
];

# ╔═╡ 94628eb5-9ee7-48db-b54b-6d352657796f
@bind pick Select(["$i" for i in 1:length(zoo)])

# ╔═╡ 23f92eab-c02a-4dcf-99a7-d170183cc67a
A = zoo[parse(Int,pick)]

# ╔═╡ 7cc1134c-e969-48e0-9700-a44bb64ba11d
U, s, V = svd(A)

# ╔═╡ e5d27592-3ddb-49d0-82f1-aa6e15683e99
U * Diagonal(s) * V' # equals A, almost

# ╔═╡ f2042d60-952e-4498-8a36-d55d54a88484
begin
Φ = range(0, 2π, 181)
xc = [cos.(Φ) sin.(Φ)]' # for drawing unit circle
yc = A * xc
end;

# ╔═╡ 132a3ebd-e9bf-4982-a8bc-010e14a16926
angles = mod.(atan.(V[2,:] ./ V[1,:]), π) # start by showing right sing. vectors

# ╔═╡ 1de925ad-4b1f-4d2c-9e1e-2ba67923e2f9
  md"Show singular vectors? $(@bind show_vec CheckBox())"

# ╔═╡ 6bf0c7c4-1195-11ec-1f36-3dfdb0d21562
md"Sliders for two input angles with special initialization"

# ╔═╡ 0bd19919-e545-4450-a6e4-bf737be8eda0
@bind angle1 Slider(range(0, 2π, 181); default=angles[1])

# ╔═╡ b5134866-436b-4720-9d10-b26bbd4c2f5a
@bind angle2 Slider(range(0, 2π, 181); default=angles[2])

# ╔═╡ 6bf0c7ce-1195-11ec-1989-9710e3fd2759
begin
    x1 = [cos(angle1), sin(angle1)] # 1st vector
    x2 = [cos(angle2), sin(angle2)] # 2nd vector
	y1 = A * x1 # matrix-vector multiplication of interest
    y2 = A * x2
	
    plot(xc[1,:], xc[2,:], line=(:black),
        xlabel=L"x_1", ylabel=L"x_2",
        xlim=(-1.1, 1.1) .* norm(A), xtick=-4:4,
        ylim=(-1.1, 1.1) .* norm(A), ytick=-4:4,
        legend=:bottomright, aspect_ratio=:equal)
    plot!(yc[1,:], yc[2,:], line=(:green,:dash))
    plot!([0, x1[1]], [0, x1[2]], line=(:blue),
        markershape=:rect, markercolor=:blue, label="x1")
    plot!([0, y1[1]], [0, y1[2]], line=(:blue,:dash),
        markershape=:circle, markercolor=:blue, label="y1 = A x1")
    plot!([0, x2[1]], [0, x2[2]], line=(:red),
        markershape=:rect, markercolor=:red, label="x2")
    plot!([0, y2[1]], [0, y2[2]], line=(:red,:dash),
        markershape=:circle, markercolor=:red, label="y2 = A x2")
if show_vec
	plot!([0, V[1,1]], [0, V[2,1]], line=(:magenta,:dash),
        markershape=:dtriangle, markercolor=:magenta, label="v1")
	plot!([0, V[1,2]], [0, V[2,2]], line=(:orange,:dash),
        markershape=:utriangle, markercolor=:orange, label="v2")
	plot!([0, s[1]*U[1,1]], [0, s[1]*U[2,1]], line=(:magenta,:dot),
        markershape=:ltriangle, markercolor=:magenta, label="u1")
	plot!([0, s[2]*U[1,2]], [0, s[2]*U[2,2]], line=(:orange,:dot),
        markershape=:rtriangle, markercolor=:orange, label="u2")
end
plot!() # for html	
end

# ╔═╡ 6bf0c7d6-1195-11ec-01d7-b306f850ccf0
#savefig("tmp.pdf")

# ╔═╡ d957eb3d-a84e-40dc-868e-c01a85cec621
U[:,1]*s[1], A * V[:,1], U[:,2]*s[2], A * V[:,2]

# ╔═╡ 79629861-b47b-458f-a474-5ea26845f560
norm(A), opnorm(A)

# ╔═╡ 808ec3e9-7690-451a-97ec-5d37760c5d9a
eigen(A)

# ╔═╡ 5133ea2e-3eb6-4901-8d20-ac30d87d2457
sign(10*exp(im*π/4)), sign(0)

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
LaTeXStrings = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f"
LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"

[compat]
LaTeXStrings = "~1.3.0"
Plots = "~1.39.0"
PlutoUI = "~0.7.52"
"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
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[[deps.Unitful]]
deps = ["Dates", "LinearAlgebra", "Random"]
git-tree-sha1 = "a72d22c7e13fe2de562feda8645aa134712a87ee"
uuid = "1986cc42-f94f-5a68-af5c-568840ba703d"
version = "1.17.0"

    [deps.Unitful.extensions]
    ConstructionBaseUnitfulExt = "ConstructionBase"
    InverseFunctionsUnitfulExt = "InverseFunctions"

    [deps.Unitful.weakdeps]
    ConstructionBase = "187b0558-2788-49d3-abe0-74a17ed4e7c9"
    InverseFunctions = "3587e190-3f89-42d0-90ee-14403ec27112"

[[deps.UnitfulLatexify]]
deps = ["LaTeXStrings", "Latexify", "Unitful"]
git-tree-sha1 = "e2d817cc500e960fdbafcf988ac8436ba3208bfd"
uuid = "45397f5d-5981-4c77-b2b3-fc36d6e9b728"
version = "1.6.3"

[[deps.Unzip]]
git-tree-sha1 = "ca0969166a028236229f63514992fc073799bb78"
uuid = "41fe7b60-77ed-43a1-b4f0-825fd5a5650d"
version = "0.2.0"

[[deps.Wayland_jll]]
deps = ["Artifacts", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "ed8d92d9774b077c53e1da50fd81a36af3744c1c"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.21.0+0"

[[deps.Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4528479aa01ee1b3b4cd0e6faef0e04cf16466da"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.25.0+0"

[[deps.XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Zlib_jll"]
git-tree-sha1 = "04a51d15436a572301b5abbb9d099713327e9fc4"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.10.4+0"

[[deps.XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.34+0"

[[deps.XZ_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "cf2c7de82431ca6f39250d2fc4aacd0daa1675c0"
uuid = "ffd25f8a-64ca-5728-b0f7-c24cf3aae800"
version = "5.4.4+0"

[[deps.Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "afead5aba5aa507ad5a3bf01f58f82c8d1403495"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.8.6+0"

[[deps.Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "6035850dcc70518ca32f012e46015b9beeda49d8"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.11+0"

[[deps.Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.0+4"

[[deps.Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "34d526d318358a859d7de23da945578e8e8727b7"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.4+0"

[[deps.Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.4+4"

[[deps.Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "5.0.3+4"

[[deps.Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[deps.Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[deps.Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[deps.Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "8fdda4c692503d44d04a0603d9ac0982054635f9"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.1+0"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "b4bfde5d5b652e22b9c790ad00af08b6d042b97d"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.15.0+0"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "730eeca102434283c50ccf7d1ecdadf521a765a4"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.2+0"

[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "330f955bc41bb8f5270a369c473fc4a5a4e4d3cb"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.6+0"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "691634e5453ad362044e2ad653e79f3ee3bb98c3"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.39.0+0"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "e92a1a012a10506618f10b7047e478403a046c77"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.5.0+0"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.13+0"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "49ce682769cd5de6c72dcf1b94ed7790cd08974c"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.5+0"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "868e669ccb12ba16eaf50cb2957ee2ff61261c56"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.29.0+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.4.0+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.8.0+0"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.48.0+0"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+0"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "9ebfc140cc56e8c2156a15ceac2f0302e327ac0a"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+0"
"""

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