mit18065.jl

### A Pluto.jl notebook ###
# v0.17.3

using Markdown
using InteractiveUtils

# ╔═╡ 23e4c789-2cc8-4565-9042-02168b16beac
begin
	using PlutoUI
	using LinearAlgebra
	using Symbolics

	function pp(xs...)
		for x ∈ xs
			print(isa(x, AbstractArray) ? repr("text/plain", x) : x)
		end
		print("\n\n")
	end

	term = PlutoUI.with_terminal

	md"""
	Assignments of [MIT 18.065 - Matrix Methods in Data Analysis, Signal Processing, and Machine Learning](https://ocw.mit.edu/courses/mathematics/18-065-matrix-methods-in-data-analysis-signal-processing-and-machine-learning-spring-2018/assignments/)

	Source: [https://github.com/maxwindiff/maths](https://github.com/maxwindiff/maths)
	"""
end

# ╔═╡ 3d9413a9-32aa-439e-8a5d-aa60ab15a651
md"""
### 1. The Column Space of ``A`` Contains All Vectors ``Ax``
"""

# ╔═╡ f5334940-2e47-4301-9508-ecfc7108f334
md"""
**(1.1)** Give an example where a combination of three nonzero vectors in ``\mathbf{R}^4`` is the zero vector. Then write your example in the form ``A\mathbf{x} = \mathbf{0}``. What are the shapes of ``A`` and ``\mathbf{x}`` and ``\mathbf{0}``?
"""

# ╔═╡ 246fff04-6614-11ec-22e1-d995b65a10a2
let
	A = [1 0 1
		 0 1 1
		 0 0 0
		 0 0 0]
	x = [1
	     1
	    -1]
	z = A * x
	@assert size(A) == (4,3)
	@assert size(x) == (3,)
	@assert size(z) == (4,)
	@assert z == zero(z)
end

# ╔═╡ 96c73c4a-f62c-41d6-bcc0-13944d3fd3c7
md"""
**(1.4)** Suppose ``A`` is the 3 by 3 matrix ``\mathbf{ones}(3, 3)`` of all ones. Find two independent vectors ``\mathbf{x}`` and ``\mathbf{y}`` that solve ``A\mathbf{x}=\mathbf{0}`` and ``A\mathbf{y}=\mathbf{0}``. Write that first equation ``A\mathbf{x}=\mathbf{0}`` (with numbers) as a combination of the columns of ``A``. Why don’t I ask for a third independent vector with ``A\mathbf{z} = \mathbf{0}``?
"""

# ╔═╡ 6f2f5e4e-7843-46b5-8647-3f681cfbbae0
let
	A = ones(3, 3)
	x = [1
		-1
		0]
	@assert A * x == zero(A * x)
	y = [1
		 0
	    -1]
	@assert A * y == zero(A * y)
	# A has rank 1, so dimension of null space is 3 - 1 = 2
	@assert rank(A) == 1
	@assert size(nullspace(A), 2) == 2
end

# ╔═╡ b344b879-d30a-420d-9e8c-410ab75c75a5
md"""
**(1.9)** Suppose the column space of an ``m`` by ``n`` matrix is all of ``\mathbf{R}^3``. What can you say about ``m``? What can you say about ``n``? What can you say about the rank ``r``?
"""

# ╔═╡ 5af2ab12-84bb-4cf5-b632-ffdfce21b8d0
md"""
**Ans:**

``m`` is 3.

``n`` is at least 3.

rank ``r`` is 3.
"""

# ╔═╡ a3511d6e-c7dc-44cc-a6b3-43e7440e32cc
md"""
**(1.18)** If ``A = CR``, what are the ``CR`` factors of the matrix
```math
\begin{equation*}
\begin{bmatrix}
0 & A \\
0 & A \\
\end{bmatrix}
\end{equation*}
```
"""

# ╔═╡ 005cde91-f411-40e8-a903-01506402c2be
term() do
	C = rand(-9:9, 3, 2)
	R = rand(-9:9, 2, 3)
	A = C * R
	A2 = kron([0 1
		       0 1], A)
    C2 = kron([1
	           1], C)
	R2 = kron([0 1], R)
	@assert A2 == C2 * R2

	pp("C2:\n", C2)
	pp("R2:\n", R2)
	pp("A2:\n", A2)
end

# ╔═╡ df291cf8-beb9-4481-bf71-0cc67a6dab7a
md"""
### 2. Multiplying and Factoring Matrices
"""

# ╔═╡ d9ca6fbe-dffb-4c5e-b1cf-0c80c7552088
md"""
**(2.2)** Suppose ``\mathbf{a}`` and ``\mathbf{b}`` are column vectors with components ``a_1,...,a_m`` and ``b_1,...,b_p``. Can you multiply ``\mathbf{a}`` times ``\mathbf{b}^\mathsf{T}`` (yes or no)? What is the shape of the answer ``\mathbf{ab}^\mathsf{T}``? What number is in row ``i``, column ``j`` of ``\mathbf{ab}^\mathsf{T}``? What can you say about ``\mathbf{aa}^\mathsf{T}``?
"""

# ╔═╡ d4f90e59-3b6b-4b51-b1ae-5c0f2b2aa8ea
term() do
	m = 3
	p = 5
	a = rand(-9:9, m, 1)
	b = rand(-9:9, p, 1)
	abT = a * transpose(b)
	pp("a:\n", a)
	pp("b:\n", b)
	pp("abT:\n", abT)

	@assert size(abT) == (m, p)

	i = rand(1:m)
	j = rand(1:p)
	@assert abT[i,j] == a[i] * b[j]

	@assert issymmetric(a * transpose(a))
end

# ╔═╡ ced55756-9423-4f66-a755-fa7afb61c159
md"""
**(2.6)** If ``A`` has columns ``\mathbf{a}_1``, ``\mathbf{a}_2``, ``\mathbf{a}_3`` and ``B = I`` is the identity matrix, what are the rank one matrices ``\mathbf{a}_1\mathbf{b}^*_1`` and ``\mathbf{a}_2\mathbf{b}^*_2`` and ``\mathbf{a}_3\mathbf{b}^*_3``? They should add to ``AI = A``.
"""

# ╔═╡ 3d07a918-0263-4d1e-9187-b0adbbe20203
term() do
	a₁ = rand(-9:9, 5, 1)
	a₂ = rand(-9:9, 5, 1)
	a₃ = rand(-9:9, 5, 1)
	A = [a₁ a₂ a₃]
	pp("A:\n", A)

	B = I(3)
	pp("B:\n", B)

	a₁b₁T = a₁ * B[:,1]'
	a₂b₂T = a₂ * B[:,2]'
	a₃b₃T = a₃ * B[:,3]'
	@assert a₁b₁T + a₂b₂T + a₃b₃T == A

	pp("a₁:\n", a₁)
	pp("b₁':\n", B[:,1]')
	pp("a₁b₁':\n", a₁b₁T)
end

# ╔═╡ 56a8646f-b8b7-4241-acb6-db64333ca508
md"""
### 3. Orthonormal Columns In ``Q`` Give ``Q’Q = I``
"""

# ╔═╡ f6db2e64-dfb2-42bc-82f5-4937f78a647d
md"""
**(3.2)** Draw unit vectors ``\mathbf{u}`` and ``\mathbf{v}`` that are not orthogonal. Show that ``\mathbf{w} = \mathbf{v} − \mathbf{u}(\mathbf{u}^\mathsf{T}\mathbf{v})`` is orthogonal to ``\mathbf{u}`` (and add ``\mathbf{w}`` to your picture).

"""

# ╔═╡ 77b1153f-e462-4c75-934d-1766ee7dc89a
term() do
	u = normalize([1, 0])
	v = normalize([1, 2])
	w = v - u * (u' * v)
	println("u = ", u)
	println("v = ", v)
	println("u'v = ", u'*v)
	println("u(u'v) = ", u*(u'*v))
	println("w = ", w)
	println(u'*w)
end

# ╔═╡ 13690156-b8f1-4768-b306-da7312e02f42
md"""
**(3.4)** Key property of every orthogonal matrix : ``||Q\mathbf{x}||^2 = ||\mathbf{x}||^2`` for every vector ``\mathbf{x}``. More than this, show that ``(Q\mathbf{x})^\mathsf{T}(Q\mathbf{y}) = \mathbf{x}^\mathsf{T}\mathbf{y}`` for every vector ``\mathbf{x}`` and ``\mathbf{y}``. So lengths and angles are not changed by ``Q``. Computations with ``Q`` never overflow!
"""

# ╔═╡ 1c12c1de-93a6-4b0d-ab73-65a77866ad3f
md"""
**Ans:**

```math
\begin{align}
(Q\mathbf{x})^\mathsf{T}(Q\mathbf{y}) &= \mathbf{x}^\mathsf{T}Q^\mathsf{T}Q\mathbf{y} \\
&= \mathbf{x}^\mathsf{T}\mathbf{y}
\end{align}
```
"""

# ╔═╡ 9c1e24e8-5645-40a2-8710-28652b044fe5
md"""
**(3.6)** A **permutation matrix** has the same columns as the identity matrix (in some order). Explain why this permutation matrix and every permutation matrix is orthogonal:

```math
\begin{equation*}
P = 
\begin{bmatrix}
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
\end{bmatrix}
\end{equation*}
```
has orthonormal basis so ``P^\mathsf{T}P = \_\_`` and ``P^{-1} = \_\_``.

When a matrix is symmetric or orthogonal, it will have orthogonal eigenvectors. This is the most important source of orthogonal vectors in applied mathematics.
"""

# ╔═╡ 9902ff32-2f28-4685-ac4e-a7ca492afe04
md"""
**Ans:** By definition, permutation matrices are row/column permutations of the identity matrix, so rows/columns are orthogonal to each other and have norm 1.

``P^\mathsf{T}P = I``

``P^{-1} = P^\mathsf{T}``
"""

# ╔═╡ 2a3b2f9f-da40-45c9-a72e-b2a85f488981
term() do
	P = [0 1 0 0
	     0 0 1 0
	     0 0 0 1
	     1 0 0 0]
	pp(P' * P)
end

# ╔═╡ f3f2074c-ed22-4637-93f9-1f59dc95f00d
md"""
### 4. Eigenvalues and Eigenvectors
"""

# ╔═╡ 30f3bf26-c49b-450e-8679-6f1536dd6a9e
md"""
**(4.2)** Compute the eigenvalues and eigenvectors of ``A`` and ``A^{−1}``. Check the trace!

```math
\begin{equation*}
A =
\begin{bmatrix}
0 & 2 \\
1 & 1 \\
\end{bmatrix}

\ \ \textrm{and} \ \ \

A^{-1} =
\begin{bmatrix}
-1/2 & 1 \\
1/2 & 0 \\
\end{bmatrix}
\end{equation*}
```
"""

# ╔═╡ f070e8d6-3514-4f11-9167-11842fbb61ff
term() do
	A = [0 2
	     1 1]
	println("λ(A) = ", eigvals(A))
	pp(eigvecs(A))

	A⁻¹ = inv(A)
	println("λ(A⁻¹) = ", eigvals(A⁻¹))
	pp(eigvecs(A⁻¹))
end

# ╔═╡ 0d0b23de-76f4-4e95-9f15-4e3c960b2935
md"""
This is because:

```math
\begin{equation*}
\begin{aligned}
A x &= λ x \\
A^{-1}Ax &= A^{-1} λ x \\
x &= A^{-1} λ x \\
\frac{1}{λ} x &= A^{-1} x
\end{aligned}
\end{equation*}
```
"""

# ╔═╡ bb82e3a6-8783-48a3-8807-c2836140fc97
md"""
**(4.11)** The eigenvalues of ``A`` equal the eigenvalues of ``A^\mathsf{T}``. This is because ``\textrm{det}(A − λI)`` equals ``\textrm{det}(A^\mathsf{T} − λI)``. That is true because __. Show by an example that the eigenvectors of ``A`` and ``A^\mathsf{T}`` are not the same.
"""

# ╔═╡ fb7f94d5-b746-4d40-8f8e-0127da73260c
term() do
	A = [0 2
	     1 1]
	println("λ(A) = ", eigvals(A), "\n")
	pp("eigvecs(A) = ", eigvecs(A))

	println("λ(A') = ", eigvals(A'), "\n")
	pp("eigvecs(A') = ", eigvecs(A'))
end

# ╔═╡ e276d15d-c643-44fa-a64a-e221a1620d24
md"""
**(4.15)** Factor these two matrices into ``A = X Λ X^{-1}``:
"""

# ╔═╡ e31e2250-00ba-46c6-93c2-0f47b5562f1c
term() do
	A = [1 2
	     0 3]
	X = eigvecs(A)
	Λ = Diagonal(eigvals(A))
	@assert A ≈ X * Λ * inv(X)

	pp("X = ", X)
	pp("Λ = ", Λ)
end

# ╔═╡ 36767f04-0101-4188-847f-633c4b21079a
term() do
	A = [1 1
	     3 3]
	X = eigvecs(A)
	Λ = Diagonal(eigvals(A))
	@assert A ≈ X * Λ * inv(X)

	pp("X = ", X)
	pp("Λ = ", Λ)
end

# ╔═╡ c1813778-6a52-4ba8-abee-57377f1d50fa
md"""
### 5. Positive Definite and Semidefinite Matrices
"""

# ╔═╡ e1285f5d-121b-4be5-80ea-afcb7cc87536
md"""
**(5.3)** For which numbers ``b`` and ``c`` are these matrices positive definite?

```math
\begin{equation*}
S =
\begin{bmatrix}
1 & b \\
b & 9 \\
\end{bmatrix}

\ \ \ \ \ \

S =
\begin{bmatrix}
2 & 4 \\
4 & c \\
\end{bmatrix}

\ \ \ \ \ \

S =
\begin{bmatrix}
c & b \\
b & c \\
\end{bmatrix}
\end{equation*}
```

With the pivots in ``D`` and multiplier in ``L``, factor each ``A`` into ``LDL^\mathsf{T}``.
"""

# ╔═╡ 582db95d-38d9-4c79-b28e-95b4f9456c54
term() do
	@variables b c

	pp([1 0; b 1] * [1 0; 0 9-b^2] * [1 b; 0 1])

	pp([1 0; 2 1] * [2 0; 0 c-8] * [1 2; 0 1])

	s3 = [1 0; b/c 1] * [c 0; 0 c-b^2/c] * [1 b/c; 0 1]
	pp(substitute(s3, Dict(b*c/c => b)))
end

# ╔═╡ f2312f44-b8c2-40a7-9d00-612d885fa251
md"""
**(5.14)** Find the 3 by 3 matrix ``S`` and its pivots, rank, eigenvalues, and determinant:

```math
\begin{bmatrix}
x_1 & x_2 & x_3
\end{bmatrix}

\begin{bmatrix}
S
\end{bmatrix}

\begin{bmatrix}
x_1 \\ x_2 \\ x_3
\end{bmatrix}

=

4(x_1 - x_2 + 2x_3)^2
```
"""

# ╔═╡ 86cb5caf-667a-465d-970c-047c6ee0e515
term() do
	@variables x₁ x₂ x₃
	pp("RHS = ", simplify(4(x₁ - x₂ + 2x₃)^2; expand=true))

	v = [2, -2, 4]
	S = v * v'
	pp("LHS = ", simplify.([x₁ x₂ x₃] * S * [x₁, x₂, x₃]; expand=true))

	@assert rank(S) == 1  # because S is the outer product of two vectors
	@assert det(S) == 0   # because S has rank 1 (not full rank)
	@assert v' * v ∈ eigvals(S)  # because S v = (v v') v = (v'*v) v
end

# ╔═╡ 208c0647-e204-4a0b-aef9-d3b322755c95
md"""
**(5.15)** Compute the three upper left determinants of ``S`` to establish positive definiteness. Verify that their ratios give the second and third pivots.

Pivots = ratios of determinants

```math
S =
\begin{bmatrix}
2 & 2 & 0 \\
2 & 5 & 3 \\
0 & 3 & 8 \\
\end{bmatrix}
```
"""

# ╔═╡ 70b16538-c1fe-46e3-bea2-2c76ec49b912
term() do
	S = [2 2 0; 2 5 3; 0 3 8]
	pp("det1 = ", det(S[1:1,1:1]))
	pp("det2/det1 = ", det(S[1:2,1:2]) / det(S[1:1,1:1]))
	pp("det3/det2 = ", det(S[1:3,1:3]) / det(S[1:2,1:2]))

	L = [1 0 0
		 1 1 0
		 0 1 1]
	U = [2 2 0
		 0 3 3
		 0 0 5]
	@assert L * U == S
	println("pivots = ", diag(U))
end

# ╔═╡ 585396de-0dab-4860-b155-5567bd66c949
md"""
### 6. Singular Value Decomposition (SVD)
"""

# ╔═╡ 3193def8-95b0-4eda-bafa-8808374be2bd
md"""
**(6.1)** A symmetric matrix ``S = S^\mathsf{T}`` has orthonormal eigenvectors ``\mathbf{v}_1`` to ``\mathbf{v}_n``. Then any vector ``\mathbf{x}`` can be written as a combination ``\mathbf{x} = c_1\mathbf{v}_1 + ... + c_n\mathbf{v}_n``. Explain these two formulas:

```math
\mathbf{x}^\mathsf{T}\mathbf{x} = c_1^2 + ... + c_n^2
\ \ \ \ \ \ \ \ \ \ \ \ \ \
\mathbf{x}^\mathsf{T}S\mathbf{x} = λ_1 c_1^2 + ... + λ_n c_2^2
```
"""

# ╔═╡ 864dd0cd-44a9-4ef6-9471-a8dbf66cfd5d
md"""
**Ans:** Self evident.
"""

# ╔═╡ c07fde15-3c68-42cd-ba68-52fea34937f1
md"""
**(6.6)** Find the ``σ``'s and ``\mathbf{v}``'s and ``\mathbf{u}``'s in the SVD for ``A = \begin{bmatrix}3 & 4 \\ 0 & 5\end{bmatrix}``. Use equation (12).
"""

# ╔═╡ 9c4089eb-e163-4104-a867-c4320816da46
term() do
	A = [3 4
	     0 5]

	vals, V = eigen(A'A)
	Σ = Diagonal(sqrt.(vals))
	pp("A'A = ", A'A)
	pp("V = ", V)
	pp("Σ = ", Σ)

	uvals, U = eigen(A*A')
	if uvals ≉ vals
		U = circshift(U, (0, 1))  # make sure eigenvectors are in the same order
	end
	pp("AA' = ", A*A')
	pp("U = ", U)

	@assert A ≈ U * Σ * V'
end

# ╔═╡ 466b265f-134c-4da8-b728-e35561c7f7fb
md"""
**(Book p.61)** If ``S = Q Λ Q^\mathsf{T}`` is symmetric positive definite, what is its SVD?
"""

# ╔═╡ c419a67a-7fc9-4d0d-ac9d-538e8a1b287c
term() do
	Q, _ = qr(randn(3,3))
	D = Diagonal([3, 2, 1])
	S = Q * D * Q'
	U, Σ, V = svd(S)

	pp("Q = ", Q)
	pp("U = ", U)
	println("Σ = ", round.(Σ), "\n")
	pp("V = ", V)

	@assert U ≈ V
	@assert all(abs.(U ./ Q) .≈ 1)  # columns of U may be negated from Q
	@assert Diagonal(Σ) ≈ D
end

# ╔═╡ 8bac8f84-790d-4d1e-ba47-f73b19472ff6
md"""
**(Book p.61)** If ``S = Q Λ Q^\mathsf{T}`` has a negative eigenvalue (``S\mathbf{x} = -α\mathbf{x}``), what is the singular value and what are the vectors ``\mathbf{v}`` and ``\mathbf{u}``?
"""

# ╔═╡ f53215cd-79b6-4343-9730-633a7db80c56
term() do
	Q, _ = qr(randn(3,3))
	D = Diagonal([3, 2, -1])
	S = Q * D * Q'
	U, Σ, V = svd(S)

	pp("Q = ", Q)
	pp("U = ", U)
	pp("Σ = ", round.(Σ))  # singular values still positive
	pp("V = ", V)  # v₃ is negated
end

# ╔═╡ 522e7390-ff59-49c8-97ee-13ac7d85aed2
md"""
**(Book p.61)** If ``A = Q`` is an orthogonal matrix, why does every singular value equal 1?
"""

# ╔═╡ 4f503ec2-3c7b-4b1d-bf86-ab7dc27c085b
term() do
	Q, _ = qr(randn(3,3))

	# Q'Q is I, so eigenvalues are all 1.
	pp("Q = ", Q)
	pp("Q'Q = ", round.(Q'Q, digits=5))
	pp("λ(Q'Q) = ", round.(eigvals(Q'Q), digits=5))

	# one way to decompose: UΣ=QV where V=I, U=Q, Σ=I ==> QI=QI
	Σ = Diagonal(sqrt.(eigvals(Q'Q)))
	@assert Σ ≈ I
	U = Q
	V = I
	@assert Q ≈ U * Σ * V'
end

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"""

# ╔═╡ Cell order:
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# ╠═9c4089eb-e163-4104-a867-c4320816da46
# ╟─466b265f-134c-4da8-b728-e35561c7f7fb
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# ╟─00000000-0000-0000-0000-000000000002