### A Pluto.jl notebook ###
# v0.16.0

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 el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : missing
        el
    end
end

# ╔═╡ b2804866-073e-11eb-3822-71c2898c5633
using LinearAlgebra

# ╔═╡ 48ddec12-0791-11eb-3278-6769b0507672
md"""# Notebook 7 -- Math 2121, Fall 2021

An $n\times n$ matrix for $n=1$ is just a number $A = \left[\begin{array}{c} a \end{array}\right]$.

Such a matrix is invertible if and only if $a\neq 0$.

It is clear that we can "move around" in the subspace of invertible $1\times 1$ matrices within $\mathbb{R}^1$, in the sense that we can perturb the entries of $A$ slightly and still have an invertible matrix: specifically, $\left[\begin{array}{c} a + \delta \end{array}\right]$ is still invertible as long as $|\delta|$ is small. 

In particular, this is true whenever $|\delta| < |a|$.

However, it's also clear that there are the set of invertible $1\times 1$ matrices has two *connected components*: the matrices $A = \left[\begin{array}{c} a \end{array}\right]$ with $a<0$ and those with $a>0$. 

If we start with $a<0$, then there is no way to gradually perturb $a$ to get a positive value without passing through $a=0$, at which point the matrix would be non-invertible. So there is no path *that stays completely within the set of invertible matrices* connecting, say $A = \left[\begin{array}{c} -1 \end{array}\right]$ and $A = \left[\begin{array}{c} 1 \end{array}\right]$.
"""

# ╔═╡ 472aca7e-0792-11eb-0957-639fae6da505
md"""Let's investigate how this generalizes to $2\times 2$ matrices.

A $2\times 2$ matrix has four entries. Geometrically, there is no difference between the set of all $2\times 2$ matrices and the set of vectors $\mathbb{R}^4$. We have just split our four numbers into two columns rather than kept them in one.

Suppose $A = \left[\begin{array}{cc} a & b \\ c & d \end{array}\right]$.

This matrix is invertible if and only if the vectors $\left[\begin{array}{c} a \\ c \end{array}\right]$ and $\left[\begin{array}{c} b \\ d \end{array}\right]$
are linearly independent, that is, not scalar multiples of each other. Assume this is the case.

I claim that it is still true that we can "move around" within the subset of invertible $2\times 2$ matrices, in the sense that the perturbed matrix 

$\left[\begin{array}{cc} a + \delta_1 & b + \delta_2 \\ c + \delta_3 & d + \delta_4 \end{array}\right]$ 

is still invertible for any choice of numbers $\delta_i$ that have sufficiently small absolute value.

Let's justify this with a random example.
"""

# ╔═╡ 40b79f8e-0731-11eb-1eb4-018915642385
A = rand(-2:0.01:2, 2, 2)

# ╔═╡ bd32c7e0-0730-11eb-1c71-d9a884839001
begin	
	using PlutoUI
	using Plots
	
	max_epsilon = 0.9 * minimum([norm(A[1:2,1]) norm(A[1:2,2])])
	def_epsilon = minimum([0.5 0.5 * max_epsilon])
	epsilon_slider = @bind epsilon Slider(0:0.01:max_epsilon, default=def_epsilon, show_value=true)
	
	
	md"""ε = $(epsilon_slider)
	"""
end

# ╔═╡ 380be1a4-08c8-11eb-1654-837ec5ab9678
md"We need a way to visualize the matrix $A$. 

A simple method is to draw the vectors $\left[\begin{array}{c} a \\ c \end{array}\right]$ and $\left[\begin{array}{c} b \\ d \end{array}\right]$ that are the columns of $A$:"

# ╔═╡ 872cf14e-0793-11eb-2909-9b4ad309de82
md"""In we increase the parameter $\varepsilon$ above, then the two colored regions in this graph will grow larger. 

The blue circle surrounds the set of vectors $\left[\begin{array}{cc} a + \delta_1  \\ c + \delta_3  \end{array}\right]$ with $\sqrt{\delta_1^2 + \delta_3^2} < \varepsilon$.

The red circle surrounds the set of vectors $\left[\begin{array}{cc} b + \delta_2  \\ d + \delta_4  \end{array}\right]$ with $\sqrt{\delta_2^2 + \delta_4^2} < \varepsilon$.

Choose $\epsilon$ to be small enough that the blue and red regions **do not overlap**.

Suppose we have another $2\times 2$ matrix $B$. Draw the columns of $B$ as a pair of vectors in $\mathbb{R}^2$. 

If the endpoints of these vectors are contained in the blue and red circles, **then $B$ is guaranteed to be invertible**, because its columns won't belong to the same line so will be linearly independent.

"""

# ╔═╡ de53a7d0-08c9-11eb-2b08-d9a5bbf5c75c
md"Given $A$ and $\varepsilon$, how would you figure out if the blue and red regions overlap? It's possible to give an exact formula for this, using only basic algebra and planar geometry. But this is a little complicated, as you can see if you examine the code below, which implements an `overlaps` method to compute the answer programatically."

# ╔═╡ 1185730a-0739-11eb-3619-1b4ee5874f4a
begin
	function tangent_points(u1, u2, eps)
		if u1 != 0
			k = -eps^2 + u1^2 + u2^2
			@assert k >= 0
			
			a, b, c = u1^2 + u2^2, -2 * u2 * k, k^2 - k * u1^2
			
			discr = b^2 - 4 * a * c
			if abs(discr) < 10e-8
				discr = 0.0
			end
			
			x2 = (-b + sqrt(discr)) / 2 / a
			x1 = -u2 / u1 * x2 + k / u1
			
			y2 = (-b - sqrt(discr)) / 2 / a
			y1 = -u2 / u1 * y2 + k / u1
			
			return x1, x2, y1, y2
		else
			x1, x2, y1, y2= tangent_points(u2, u1, eps)
			return x2, x1, y2, y1
		end
	end
	
	function angle(u1, u2)
		if u1 == 0
			return u2 > 0 ? pi/2 : 3 * pi/2
		end
		if u2 == 0
			return u1 > 0 ? 0 : pi
		end
		
		a = atan(abs(u2) / abs(u1))
		if u1 > 0 && u2 > 0
			return a
		elseif u1 > 0 && u1 > 0
			return 2 * pi - a
		elseif u1 < 0 && u2 > 0
			return pi - a
		else
			return pi + a
		end
	end
	
	function in_same_relative_position(x, y, test, target)
		a = angle(x[1], x[2])
		b = angle(y[1], y[2])
		if a > b
			a, b = b, a
		end
		
		s = angle(test[1], test[2])
		if b < s
			a, b = b, a + 2 * pi
		elseif s < a 
			a, b, s = b, a + 2 * pi, s + 2 * pi
		end
		@assert a <= s <= b
		
		t = angle(target[1], target[2])
		return a < t < b || a < t + 2 * pi < b
	end
		
	function overlaps(eps, A)
		for i = [1, 2]
		for s = [-1,1]
			u1, u2 = A[1, i], A[2, i]
			v1, v2 = s * A[1, 3 - i], s * A[2, 3- i]

			if -eps^2 + u1^2 + u2^2 < 0 || -eps^2 + v1^2 + v2^2 < 0
				return true
			end
				
			x1, x2, y1, y2 = tangent_points(u1, u2, eps)
			p1, p2, q1, q2 = tangent_points(v1, v2, eps)

			if in_same_relative_position([x1; x2], [y1; y2], [u1; u2], [p1; p2])
				return true
			end

			if in_same_relative_position([x1; x2], [y1; y2], [u1; u2], [q1; q2])
				return true
			end
		end
		end
		return false
	end
end

# ╔═╡ fd6fa06e-0730-11eb-229e-21f0a63e8feb
begin		
	mm = 1.5 * maximum(abs.(A))
	lims = (-mm, mm)
	
	rectangle(w, h, x, y) = Shape(x .+ [0,w,w,0], y .+ [0,0,h,h])

	function circle(h, k, r)
		theta = LinRange(0, 2 * pi, 500)
		h .+ r*sin.(theta), k .+ r*cos.(theta)
	end
	
	function draw(u1, u2, epsilon, c)
		N = 1000
		
		quiver!(quiver = ([u1],[u2]), [0], [0], color=[:black],)
		plot!(circle(u1, u2, epsilon), opacity=.2, color=c)

		x1, x2, y1, y2 = tangent_points(u1, u2, epsilon)
		x1, x2, y1, y2 = N * x1, N * x2, N * y1, N * y2

		plot!(Shape([0, x1, y1], [0, x2, y2]), opacity=.05, color=c)
		plot!(Shape([0, -x1, -y1], [0, -x2, -y2]), opacity=.05, color=c)
	end
		
	p1 = scatter([0], [0], xlim = lims, ylim = lims,
		legend = false, label = "origin",  aspect_ratio=:equal,
		title = "Columns of A")
	
	u1, u2 = A[1, 1], A[2, 1]
	v1, v2 = A[1, 2], A[2, 2]
	
	draw(u1, u2, epsilon, [:blue])
	draw(v1, v2, epsilon, [:red])
	
	plot(p1)
end

# ╔═╡ 11c57a80-073a-11eb-321d-578078ffcdd4
overlaps(epsilon, A)

# ╔═╡ 9b8f0c96-0793-11eb-207b-cd58e584a208
md"As long as the red and blue regions do not overlap, then  

$B=A + \left[\begin{array}{cc} \delta_1 &  \delta_2 \\ \delta_3 &  \delta_4 \end{array}\right] = \left[\begin{array}{cc} a + \delta_1 & b + \delta_2 \\ c + \delta_3 & d + \delta_4 \end{array}\right]$

is invertible for every choice $\delta_1,\delta_2,\delta_3,\delta_4$ with $\sqrt{\delta_1^2 + \delta_3^2} < \varepsilon$ and $\sqrt{\delta_2^2 + \delta_4^2} < \varepsilon$. 

These conditions involving square roots are satisfied as long as we have 

$-\frac{1}{\sqrt {2}}\varepsilon < \delta_i< \frac{1}{\sqrt 2}\varepsilon$

for each $i=1,2,3,4$.

This shows that we can 'move' from $A$ to any matrix $B$ whose columns correspond to points in the red and blue circles, while always staying inside the set of invertible $2\times 2$ matrices.

The following method computes an upper bound on the value of $\varepsilon$ that ensures that the red and blue regions above do not overlap. I'll call this the *radius* of $A$."

# ╔═╡ d481fb8a-073e-11eb-15a4-9be4957b67a3
function radius(A, reduce_scale=1.0)
	ans = 1.0 # start with a guess of 1.1
	factor = 1.1 # factor to adjust by at each iteration
	
	iter, limit = 0, 100 # parameters to bound number of iterations
	while true
		over = overlaps(ans, A)
		overup = overlaps(factor * ans, A)
		
		# if current answer works and can't do better, break
		if ! over && (overup || iter == limit)
			break
			
		# if current answer works but can do better, increase value
		elseif ! over && ! overup
			ans *= factor
			
		# if current answer doesn't work, decrease value
		else
			ans /= factor
		end
		
		# quite if already spent too many iterations searching
		iter += 1
		if iter > limit
			return 0.0
		end
	end
	
	# perfect extra size reduction, just in case.
	# these formulas will make more sense after we learn about projections.
	u, v = A[1:2,1], A[1:2,2]
	n1 = norm(v - dot(u, v) / dot(u, u) * u)
	n2 = norm(u - dot(u, v) / dot(v, v) * v)
	ans = minimum([n1 n2 ans])
	
	# optional size reduction
	ans = reduce_scale * ans
	
	# reduce answer to 0.0 if it's very small, to avoid roundoff errors
	ans = ans < 0.001 ? 0.0 : ans 
	
	return ans
end

# ╔═╡ 30818900-073f-11eb-3f9c-f76873689dc7
radius(A)

# ╔═╡ b77c6bfb-f422-4903-a52a-1c71899b10c2
overlaps(radius(A), A)

# ╔═╡ 4feb3f2a-07ad-11eb-2ed0-216ffc8a5494
md"Let's create another random invertible 2-by-2 matrix $B$."

# ╔═╡ fc2a847a-0740-11eb-2034-9f05722e243e
begin
	B = rand(-2:0.01:2, 2, 2)
	[A, B]
end

# ╔═╡ 418e2d42-0786-11eb-295a-29facc97fbf8
begin
	bound = 1.5 * maximum([maximum(abs.(A)) maximum(abs.(B))])

	r1 = radius(A, 0.5)
	plt_A = scatter(
			[0], [0], xlim=(-bound,bound), ylim=(-bound,bound),
			legend = false, label = "origin",
			aspect_ratio=:equal,
			title = "Columns of A",
			xlabel = string("radius = ", r1))
	draw(A[1,1], A[2,1], r1, [:blue])
	draw(A[1,2], A[2,2], r1, [:red])
	
	r2 = radius(B, 0.6)
	plt_B = scatter(
			[0], [0], xlim=(-bound,bound), ylim=(-bound,bound),
			legend = false, label = "origin",
			aspect_ratio=:equal,
			title = "Columns of B",
			xlabel = string("radius = ", r2))
	draw(B[1,1], B[2,1], r2, [:blue])
	draw(B[1,2], B[2,2], r2, [:red])
	
	plot(plt_A, plt_B, layout=(1,2))
end

# ╔═╡ 681678f2-08cb-11eb-3b9c-3df9fa6aca4c
md"Is it possible to traverse a path from $A$ to $B$ that stays completely inside the set of invertible 2-by-2 matrices? 
There is certainly a path from $A$ to $B$ inside the set of *all* 2-by-2 matrices.

The simplest such path is the *line* from $A$ to $B$: this is the sequence of matrices 

$A + t(B-A)$ 

as $t \in \mathbb{R}$ varies over the numbers in the interval $0\leq t \leq 1$."

# ╔═╡ f6b0eb56-08cb-11eb-0f13-8f74518495ca
md"For our matrices, give equally spaces points on this line are given by"

# ╔═╡ 1f41c914-08cc-11eb-2a52-f7feb5ce49f8
[A + t * (B-A) for t=0:0.25:1]

# ╔═╡ 39d3040a-08cc-11eb-3612-6da902c3eec2
md"Some questions we could ask: what does this line look like in terms of our pictures of 2-by-2 matrices as pairs of vectors in $\mathbb{R}^2$? And does this line stay in the set of *invertible* 2-by-2 matrices?"

# ╔═╡ c5017644-07a8-11eb-33e9-5f665fef5bbe
begin
	# This struct represents the line from A to B described above.
	# The value of A is our current position on this line.
	Base.@kwdef mutable struct InvertibleMatrixPath
        A::Array{Float64,2} = A
		B::Array{Float64,2} = B
		
        prev_A::Array{Array{Float64,2},1} = []
		prev_r::Array{Float64,1} = []
    end
    
	# Every time the `step` method is called, we move forward on the line to B.
	# The catch is that we only move the distance that is guaranteed to keep
	# us inside the set of invertible matrices. This means that the distance
	# we travel could become arbitrary small; we could get stuck on the path to B.
    function step!(ch::InvertibleMatrixPath)
		r = radius(ch.A, 0.5)
		
		push!(ch.prev_r, r)
		push!(ch.prev_A, ch.A)
		
		diff = ch.B - ch.A
		r = norm(diff) < r ? 1.0 : r / norm(diff)
		ch.A += diff * r
    end
end

# ╔═╡ 47fb9168-08cd-11eb-1bdc-eb832e330f3c
md"Using the above struct, we can implement a method that tells us whether the line from A to B does stay inside the set of invertible 2-by-2 matrices."

# ╔═╡ 8d7389d2-07a9-11eb-1355-996acad905a3
function line_stays_invertible(A, B)
	line = InvertibleMatrixPath(A, B, [], [])
	
	while true
		step!(line)
		if norm(line.A - B) < 10e-8 # we have reached B successfully
			return true
		end
		if line.prev_r[end] == 0 # we have gotten stuck on our path
			return false
		end
	end
end	

# ╔═╡ 796fd045-f5a6-4bfb-9f0e-b12b9c83ef3b
md"The following method returns how many steps we take before we know if our line reaches $B$."

# ╔═╡ 1fbea058-3a08-48f9-aa50-691dec0a78f7
function path_length(A, B; step_fn=step!)
    path = InvertibleMatrixPath(A, B, [], [])
    step_count = 1
    while true
        step_fn(path)
        if path.prev_r[end] == 0 || norm(path.A - B) < 10e-8
            break
        end
        step_count += 1
    end
    return step_count
end

# ╔═╡ 3ef26f56-0742-11eb-26ef-fb5c86b13ea5
md"For the matrices $A$ and  $B$:"

# ╔═╡ df27ca0c-08cd-11eb-2db7-45151197da1d
[A, B]

# ╔═╡ eb62367e-07a9-11eb-09be-952eb527f1f8
line_stays_invertible(A, B)

# ╔═╡ f019ae84-08cd-11eb-11ea-67eb238b9b7b
md"The animation below shows the path that results from trying to follow the line from $A$ to $B$ while staying inside the set of invertible matrices."

# ╔═╡ c57aad40-0783-11eb-3e0f-a54699def5dd
function animation(A, B; step_fn=step!)
	bound = 1.5 * maximum([maximum(abs.(A)) maximum(abs.(B))])
	path = InvertibleMatrixPath(A, B, [], [])	
	steps = 1.25 * path_length(A, B; step_fn=step_fn)
	anim = @animate for j=1:steps
		e = radius(path.A, 0.5)
		plt = scatter(
			[0], [0], xlim=(-bound,bound), ylim=(-bound,bound),
			legend = false, label = "origin",
			aspect_ratio=:equal,
			title = "Traversing a path from A to B in space of 2-by-2 matrices",
			ylabel = line_stays_invertible(A, B) ? "outcome: line connects" : "outcome: line is obstructed",
			xlabel = string("radius = ", e)
		)

		for i=1:length(path.prev_r)
			plot!(circle(path.prev_A[i][1,1], path.prev_A[i][2,1], path.prev_r[i]), opacity=.15, fillalpha=0.2, color=[:blue])
			plot!(circle(path.prev_A[i][1,2], path.prev_A[i][2,2], path.prev_r[i]), opacity=.15, color=[:red])
		end

		draw(path.A[1,1], path.A[2,1], e, [:blue])
		draw(path.A[1,2], path.A[2,2], e, [:red]) 

		quiver!(quiver = ([B[1,1]],[B[2,1]]), [0], [0], color = [:grey],)
		quiver!(quiver = ([B[1,2]],[B[2,2]]), [0], [0], color = [:grey],)

		step_fn(path) 
	end every 1
	gif(anim, "anim_fps15.gif", fps = 5)
end

# ╔═╡ b1f86522-08d3-11eb-0e0d-c18c9f382198
animation(A, B)

# ╔═╡ e329735e-074b-11eb-236d-e5f04a4ffa81
md"Sometimes the line from $A$ to $B$ stays in the set of $2\times 2$ invertible matrices, sometimes it doesn't.

A region of space that contains the line segment connecting any two of its points is called *convex*. The solid sphere is convex. The solid torus (donut shape) is not convex. All regular polygons are convex regions of $\mathbb{R}^2$. Star shapes are not convex.

Our experiments show that the set of invertible 2-by-2 matrices is not convex."

# ╔═╡ 17923f78-e1cd-4bde-b259-07447df8bea3
md"We can actually predict exactly when the line between two invertible 2-by-2 matrices stays completely inside the set of invertible matrices. 

There is an easily detected case in which this can never happen."

# ╔═╡ 0dfd0afc-08d0-11eb-1b68-333d24004038
md"We saw in the lecture notes that $A = \left[\begin{array}{cc} a & b \\ c & d \end{array}\right]$ is invertible if and only if $ad - bc \neq 0$.

Next week, we'll see that the number $ad-bc$ is called the *determinant* of $A$, and is written $\det A$."

# ╔═╡ b666ae90-4550-459d-88bf-27f8cbf60d2c
A

# ╔═╡ a29e1804-8ed1-4a43-94f7-b8b831187019
A[1, 1] * A[2, 2] - A[1, 2] * A[2, 1]

# ╔═╡ 9c196c18-1a9f-482d-8c76-2b4106d6245b
# computing the determinant using a built-in function
det(A)

# ╔═╡ f4b117ea-08d0-11eb-3bbb-192bd2462382
md"When we perturb $A$ slightly, the value of $\det A$ changes by a small amount. So if $\det A$ starts out positive, and we perform a sequence of perturbations that remain in the set of invertible matrices, ending up at some invertible matrix $B$, then $\det B$ must also be positive.

Traveling along our invertible matrix path from $A$ to $B$ cannot change the sign of the determinant. We can only change the sign by passing through a non-invertible matrix."

# ╔═╡ c6c1d7a8-a49f-4f51-af18-2a53832ed11d
function find_matrices_with_determinants_of_opposite_sign(range)
	while true	
		A = rand(range, 2, 2)
		B = rand(range, 2, 2)

		if det(A) * det(B) >= 0
			continue
		end

		return A, B
	end
end

# ╔═╡ 0b42bc8e-42d1-49ac-a2de-70bddd3afe70
X, Y = find_matrices_with_determinants_of_opposite_sign(-10:0.1:10)

# ╔═╡ 536e829b-0be7-48d8-b3ce-fb03b98e3845
[det(X), det(Y)]

# ╔═╡ 142a4e5f-1d51-45d6-9385-37b96fe45c12
line_stays_invertible(X, Y)

# ╔═╡ 7dd60a62-08d1-11eb-05fa-0dc606c07fd9
md"It turns out that any two 2-by-2 matrices whose determinants are both positive or both negative are connected by a path the stays completely inside the set of invertible matrices. 

However, the line connecting the two matrices may fail to be such a path: the subset of 2-by-2 invertible matrices with positive (or negative) determinant is also not convex. We can check this:"

# ╔═╡ a2ff113c-4458-4c73-8a84-1e614829055e
function find_reachable(range; samesign=false)
	while true	
		A = rand(range, 2, 2)
		B = rand(range, 2, 2)
		
		if samesign && det(A) * det(B) < 0
			continue
		end
		
		if line_stays_invertible(A, B)
			return A, B
		end
	end
end

# ╔═╡ 24fc4f6a-08c6-11eb-3c8c-454186761f3b
function find_unreachable(range; samesign=false)
	while true	
		A = rand(range, 2, 2)
		B = rand(range, 2, 2)
		
		if samesign && det(A) * det(B) < 0
			continue
		end
		
		if ! line_stays_invertible(A, B)
			return A, B
		end
	end
end

# ╔═╡ 6b893fb8-08d2-11eb-0238-216af1426550
md"Here are two matrices with determinant of the same sign:"

# ╔═╡ cb1084de-08d0-11eb-0ec3-c7daba1f8d83
source, target = find_unreachable(-10:0.01:10; samesign=true)

# ╔═╡ 330bcc92-08d1-11eb-136a-7ff0d5615523
[det(source) det(target)]

# ╔═╡ 7c18961c-08d2-11eb-15db-4fa7ee06e328
md"However, the line between them does not stay in the set of invertible matrices:"

# ╔═╡ 3e7a65c6-08c6-11eb-2ba3-a30de3f58923
line_stays_invertible(source, target)

# ╔═╡ 10ad04f2-e9ec-4d9a-a96b-dd4b2f6f529d
path_length(source, target)

# ╔═╡ 386c38e4-08c6-11eb-03d3-8f4d9e50f13e
animation(source, target)

# ╔═╡ f8debe6d-5669-4f78-8eb4-31ed4d77ebea
md"So if we have two matrices whose determinants are both positive or both negative, how can we tell if every point on the line connecting them in the space of matrices is also invertible?

The *trace* of a matrix $A$ is the sum of its diagonal entries."

# ╔═╡ 6bfcf870-69fc-4953-a4c1-6fdc9fe20cd4
A

# ╔═╡ d90d0106-2577-4600-986f-d7bf259ca6ec
A[1, 1] + A[2, 2]

# ╔═╡ 7b43ea16-8cda-464b-96c3-cabe79041ee4
# computing the trace using a built-in function
tr(A)

# ╔═╡ 20003261-52eb-4845-9864-87dbc2d1d7ab
md"Assume $A$ and $B$ are 2-by-2 matrices whose determinants are both positive or both negative. 

Let $M = AB^{-1}$. It turns out that every point on the line from $A$ to $B$ is invertible if and only if

$\det(M)\mathrm{tr}(M) >0 \quad\text{or}\quad 4 \det(M) - \mathrm{tr}(M)^2 > 0.$
"

# ╔═╡ 27ef7cfd-f044-45a4-acdb-5a3c44855649
function predict_line_stays_invertible(source, target)
	if det(source) * det(target) <= 0
		return false
	end
	
	M = source * target^-1
	
	if tr(M) * det(M) > 0
		return true
	end
	
	if 4 * det(M) - tr(M)^2 > 0
		return true
	end
	
	return false
end

# ╔═╡ 8c991270-cc88-41b6-9e45-f85ebefecd21
S1, T1 = find_unreachable(-10:0.01:10)

# ╔═╡ b08af805-6e0a-49d6-9789-7b1eca647d41
line_stays_invertible(S1, T1)

# ╔═╡ 3169b2f9-9ac5-4dfc-8f6e-745bf1a9134d
predict_line_stays_invertible(S1, T1)

# ╔═╡ 38ac1249-5b50-4792-bb27-c7e87bf8a1f1
S2, T2 = find_reachable(-10:0.01:10)

# ╔═╡ fa41b995-9c1d-4b18-8b4c-d75ba0d52e49
line_stays_invertible(S2, T2)

# ╔═╡ 22710f6d-448f-4a34-856b-5c6610dae3d8
predict_line_stays_invertible(S2, T2)

# ╔═╡ 8b623ae2-f8d2-4b39-b8d5-d1b24be4a363
md"We will be able to justify why works later in the course. For now, it's a bit of a mystery. 

After the midterm, we will talk about the *eigenvalues* of a matrix. It turns out that the conditions above are equivalent to requiring that $AB^{-1}$ have **no real negative eigenvalues**.

When $n>2$, here is a good guess for when the line connecting two invertible $n$-by-$n$ matrices $A$ and $B$ consists entirely of invertible matrices: exactly when $\det(A)$ and $\det(B)$ are both positive or both negative, and $AB^{-1}$ has no real negative eigenvalues."

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

[compat]
Plots = "~1.22.3"
PlutoUI = "~0.7.14"
"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
# This file is machine-generated - editing it directly is not advised

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[[GLFW_jll]]
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[[Libtiff_jll]]
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[[Libuuid_jll]]
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uuid = "d6f4376e-aef5-505a-96c1-9c027394607a"

[[MbedTLS]]
deps = ["Dates", "MbedTLS_jll", "Random", "Sockets"]
git-tree-sha1 = "1c38e51c3d08ef2278062ebceade0e46cefc96fe"
uuid = "739be429-bea8-5141-9913-cc70e7f3736d"
version = "1.0.3"

[[MbedTLS_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "c8ffd9c3-330d-5841-b78e-0817d7145fa1"

[[Measures]]
git-tree-sha1 = "e498ddeee6f9fdb4551ce855a46f54dbd900245f"
uuid = "442fdcdd-2543-5da2-b0f3-8c86c306513e"
version = "0.3.1"

[[Missings]]
deps = ["DataAPI"]
git-tree-sha1 = "bf210ce90b6c9eed32d25dbcae1ebc565df2687f"
uuid = "e1d29d7a-bbdc-5cf2-9ac0-f12de2c33e28"
version = "1.0.2"

[[Mmap]]
uuid = "a63ad114-7e13-5084-954f-fe012c677804"

[[MozillaCACerts_jll]]
uuid = "14a3606d-f60d-562e-9121-12d972cd8159"

[[NaNMath]]
git-tree-sha1 = "bfe47e760d60b82b66b61d2d44128b62e3a369fb"
uuid = "77ba4419-2d1f-58cd-9bb1-8ffee604a2e3"
version = "0.3.5"

[[NetworkOptions]]
uuid = "ca575930-c2e3-43a9-ace4-1e988b2c1908"

[[Ogg_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "7937eda4681660b4d6aeeecc2f7e1c81c8ee4e2f"
uuid = "e7412a2a-1a6e-54c0-be00-318e2571c051"
version = "1.3.5+0"

[[OpenSSL_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "15003dcb7d8db3c6c857fda14891a539a8f2705a"
uuid = "458c3c95-2e84-50aa-8efc-19380b2a3a95"
version = "1.1.10+0"

[[Opus_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "51a08fb14ec28da2ec7a927c4337e4332c2a4720"
uuid = "91d4177d-7536-5919-b921-800302f37372"
version = "1.3.2+0"

[[OrderedCollections]]
git-tree-sha1 = "85f8e6578bf1f9ee0d11e7bb1b1456435479d47c"
uuid = "bac558e1-5e72-5ebc-8fee-abe8a469f55d"
version = "1.4.1"

[[PCRE_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "b2a7af664e098055a7529ad1a900ded962bca488"
uuid = "2f80f16e-611a-54ab-bc61-aa92de5b98fc"
version = "8.44.0+0"

[[Parsers]]
deps = ["Dates"]
git-tree-sha1 = "a8709b968a1ea6abc2dc1967cb1db6ac9a00dfb6"
uuid = "69de0a69-1ddd-5017-9359-2bf0b02dc9f0"
version = "2.0.5"

[[Pixman_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "b4f5d02549a10e20780a24fce72bea96b6329e29"
uuid = "30392449-352a-5448-841d-b1acce4e97dc"
version = "0.40.1+0"

[[Pkg]]
deps = ["Artifacts", "Dates", "Downloads", "LibGit2", "Libdl", "Logging", "Markdown", "Printf", "REPL", "Random", "SHA", "Serialization", "TOML", "Tar", "UUIDs", "p7zip_jll"]
uuid = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f"

[[PlotThemes]]
deps = ["PlotUtils", "Requires", "Statistics"]
git-tree-sha1 = "a3a964ce9dc7898193536002a6dd892b1b5a6f1d"
uuid = "ccf2f8ad-2431-5c83-bf29-c5338b663b6a"
version = "2.0.1"

[[PlotUtils]]
deps = ["ColorSchemes", "Colors", "Dates", "Printf", "Random", "Reexport", "Statistics"]
git-tree-sha1 = "b084324b4af5a438cd63619fd006614b3b20b87b"
uuid = "995b91a9-d308-5afd-9ec6-746e21dbc043"
version = "1.0.15"

[[Plots]]
deps = ["Base64", "Contour", "Dates", "Downloads", "FFMPEG", "FixedPointNumbers", "GR", "GeometryBasics", "JSON", "Latexify", "LinearAlgebra", "Measures", "NaNMath", "PlotThemes", "PlotUtils", "Printf", "REPL", "Random", "RecipesBase", "RecipesPipeline", "Reexport", "Requires", "Scratch", "Showoff", "SparseArrays", "Statistics", "StatsBase", "UUIDs"]
git-tree-sha1 = "cfbd033def161db9494f86c5d18fbf874e09e514"
uuid = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
version = "1.22.3"

[[PlutoUI]]
deps = ["Base64", "Dates", "HypertextLiteral", "IOCapture", "InteractiveUtils", "JSON", "Logging", "Markdown", "Random", "Reexport", "UUIDs"]
git-tree-sha1 = "d1fb76655a95bf6ea4348d7197b22e889a4375f4"
uuid = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
version = "0.7.14"

[[Preferences]]
deps = ["TOML"]
git-tree-sha1 = "00cfd92944ca9c760982747e9a1d0d5d86ab1e5a"
uuid = "21216c6a-2e73-6563-6e65-726566657250"
version = "1.2.2"

[[Printf]]
deps = ["Unicode"]
uuid = "de0858da-6303-5e67-8744-51eddeeeb8d7"

[[Qt5Base_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "Fontconfig_jll", "Glib_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "OpenSSL_jll", "Pkg", "Xorg_libXext_jll", "Xorg_libxcb_jll", "Xorg_xcb_util_image_jll", "Xorg_xcb_util_keysyms_jll", "Xorg_xcb_util_renderutil_jll", "Xorg_xcb_util_wm_jll", "Zlib_jll", "xkbcommon_jll"]
git-tree-sha1 = "ad368663a5e20dbb8d6dc2fddeefe4dae0781ae8"
uuid = "ea2cea3b-5b76-57ae-a6ef-0a8af62496e1"
version = "5.15.3+0"

[[REPL]]
deps = ["InteractiveUtils", "Markdown", "Sockets", "Unicode"]
uuid = "3fa0cd96-eef1-5676-8a61-b3b8758bbffb"

[[Random]]
deps = ["Serialization"]
uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"

[[RecipesBase]]
git-tree-sha1 = "44a75aa7a527910ee3d1751d1f0e4148698add9e"
uuid = "3cdcf5f2-1ef4-517c-9805-6587b60abb01"
version = "1.1.2"

[[RecipesPipeline]]
deps = ["Dates", "NaNMath", "PlotUtils", "RecipesBase"]
git-tree-sha1 = "7ad0dfa8d03b7bcf8c597f59f5292801730c55b8"
uuid = "01d81517-befc-4cb6-b9ec-a95719d0359c"
version = "0.4.1"

[[Reexport]]
git-tree-sha1 = "45e428421666073eab6f2da5c9d310d99bb12f9b"
uuid = "189a3867-3050-52da-a836-e630ba90ab69"
version = "1.2.2"

[[Requires]]
deps = ["UUIDs"]
git-tree-sha1 = "4036a3bd08ac7e968e27c203d45f5fff15020621"
uuid = "ae029012-a4dd-5104-9daa-d747884805df"
version = "1.1.3"

[[SHA]]
uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce"

[[Scratch]]
deps = ["Dates"]
git-tree-sha1 = "0b4b7f1393cff97c33891da2a0bf69c6ed241fda"
uuid = "6c6a2e73-6563-6170-7368-637461726353"
version = "1.1.0"

[[Serialization]]
uuid = "9e88b42a-f829-5b0c-bbe9-9e923198166b"

[[SharedArrays]]
deps = ["Distributed", "Mmap", "Random", "Serialization"]
uuid = "1a1011a3-84de-559e-8e89-a11a2f7dc383"

[[Showoff]]
deps = ["Dates", "Grisu"]
git-tree-sha1 = "91eddf657aca81df9ae6ceb20b959ae5653ad1de"
uuid = "992d4aef-0814-514b-bc4d-f2e9a6c4116f"
version = "1.0.3"

[[Sockets]]
uuid = "6462fe0b-24de-5631-8697-dd941f90decc"

[[SortingAlgorithms]]
deps = ["DataStructures"]
git-tree-sha1 = "b3363d7460f7d098ca0912c69b082f75625d7508"
uuid = "a2af1166-a08f-5f64-846c-94a0d3cef48c"
version = "1.0.1"

[[SparseArrays]]
deps = ["LinearAlgebra", "Random"]
uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"

[[StaticArrays]]
deps = ["LinearAlgebra", "Random", "Statistics"]
git-tree-sha1 = "3c76dde64d03699e074ac02eb2e8ba8254d428da"
uuid = "90137ffa-7385-5640-81b9-e52037218182"
version = "1.2.13"

[[Statistics]]
deps = ["LinearAlgebra", "SparseArrays"]
uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"

[[StatsAPI]]
git-tree-sha1 = "1958272568dc176a1d881acb797beb909c785510"
uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0"
version = "1.0.0"

[[StatsBase]]
deps = ["DataAPI", "DataStructures", "LinearAlgebra", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"]
git-tree-sha1 = "8cbbc098554648c84f79a463c9ff0fd277144b6c"
uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
version = "0.33.10"

[[StructArrays]]
deps = ["Adapt", "DataAPI", "StaticArrays", "Tables"]
git-tree-sha1 = "2ce41e0d042c60ecd131e9fb7154a3bfadbf50d3"
uuid = "09ab397b-f2b6-538f-b94a-2f83cf4a842a"
version = "0.6.3"

[[TOML]]
deps = ["Dates"]
uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76"

[[TableTraits]]
deps = ["IteratorInterfaceExtensions"]
git-tree-sha1 = "c06b2f539df1c6efa794486abfb6ed2022561a39"
uuid = "3783bdb8-4a98-5b6b-af9a-565f29a5fe9c"
version = "1.0.1"

[[Tables]]
deps = ["DataAPI", "DataValueInterfaces", "IteratorInterfaceExtensions", "LinearAlgebra", "TableTraits", "Test"]
git-tree-sha1 = "fed34d0e71b91734bf0a7e10eb1bb05296ddbcd0"
uuid = "bd369af6-aec1-5ad0-b16a-f7cc5008161c"
version = "1.6.0"

[[Tar]]
deps = ["ArgTools", "SHA"]
uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e"

[[Test]]
deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"

[[URIs]]
git-tree-sha1 = "97bbe755a53fe859669cd907f2d96aee8d2c1355"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.3.0"

[[UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"

[[Wayland_jll]]
deps = ["Artifacts", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "3e61f0b86f90dacb0bc0e73a0c5a83f6a8636e23"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.19.0+0"

[[Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll"]
git-tree-sha1 = "2839f1c1296940218e35df0bbb220f2a79686670"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.18.0+4"

[[XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "1acf5bdf07aa0907e0a37d3718bb88d4b687b74a"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.9.12+0"

[[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"

[[Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "5be649d550f3f4b95308bf0183b82e2582876527"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.6.9+4"

[[Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4e490d5c960c314f33885790ed410ff3a94ce67e"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.9+4"

[[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"

[[Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fe47bd2247248125c428978740e18a681372dd4"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.3+4"

[[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"

[[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"

[[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"

[[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"

[[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"

[[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"

[[Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.0+3"

[[Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.13.0+3"

[[Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.0+4"

[[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"

[[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"

[[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"

[[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"

[[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"

[[Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"

[[Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "cc4bf3fdde8b7e3e9fa0351bdeedba1cf3b7f6e6"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.0+0"

[[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"

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

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

[[libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "c45f4e40e7aafe9d086379e5578947ec8b95a8fb"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+0"

[[nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"

[[p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"

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

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

[[xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

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