Notebook 10 – Math 2121, Fall 2020

In this short demonstration we explore the physical meaning of the sign of a determinant.

Some terminology:

An invertible matrix with positive determinant is called orientation-preserving.
An invertible matrix with negative determinant is called orientation-reversing.

Whether an $n \times n$ matrix is orientation-preserving or orientation-reversing is something we can "just see," at least when $n = 2$ , simply by looking at how the matrix transforms a set of points.

However, it's not so easy to make a rigorous definition of these properties without referencing the determinant.

15.9 μs

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using Images, FileIO, Plots, LinearAlgebra, Random

20 s

A black and white image is encoded by a set of vectors $v = [\begin{matrix} v_{1} \\ v_{2} \end{matrix}] \in R^{2}$ : one vector for each black pixel at horizontal position $v_{1}$ and vertical position $v_{2}$ .

A $2 \times 2$ matrix $A$ transforms such an image to another image, in which each vector $v$ is replaced by $A v$ . By comparing these images, we will be able to see whether $A$ is orientation-preserving or orientation-reversing.

9.2 μs

extract_image (generic function with 1 method)

 
function extract_image(filename)
    # this function converts a png file to a list of vectors as we just described
    img = download(filename) |> load
    gx, gy = [], []
    (m,n) = size(img)
    for x=1:m
        for y=1:n 
            if img[x,y] != img[1,1]
                append!(gx, x)
                append!(gy, y)
            end
        end
    end
    # now resize list of coordinates into 2-by-N matrix
    return vcat(transpose(reverse(gx)), transpose(gy))
    # note: we reverse order of x-coordinates because increasing the row
    # index goes down in a matrix, which is opposite of cartesian coordinates
end

45.1 μs

2×67870 Array{Any,2}:
 380  380  380  380  380  380  380  380  …   12   12   12   12   12   12   12   12
 198  199  200  201  202  203  204  205     431  432  433  434  435  436  437  438

 
g = extract_image("www.math.ust.hk/~emarberg/teaching/2020/Math2121/julia/base.png")

195 ms

Here is what this image looks like if we plot all of the points:

7.6 μs

make_displayable (generic function with 1 method)

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function make_displayable(g)
    # this function converts our image as list of vectors into something displayable
    padding = 3
    gx, gy = g[1,:], g[2,:]
    
    minx, miny = Int(floor(minimum(gx))) - 1, Int(floor(minimum(gy))) - 1
    maxx, maxy = Int(ceil(maximum(gx))), Int(ceil(maximum(gy)))
    
    ans = zeros(Int64, maxx - minx + padding, maxy - miny + padding)
    for i = 1:length(gx)
        x = Int(ceil(gx[i])) - minx
        y = Int(ceil(gy[i])) - miny
        for p=0:padding
            for q=0:padding
                ans[x + p, y + q] = 1
            end
        end
    end
    return ans
end

50.9 μs

134 ms

Below are some functions to create random $n$ -by- $n$ matrices with determinant $+ 1$ or $- 1$ .

We'll just need the $n = 2$ case of these methods.

11.3 μs

randmat (generic function with 1 method)

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function randmat(n)
    return 2 * rand(Float64, n, n) .- 1.0
end

48.1 μs

positive_determinant (generic function with 1 method)

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function positive_determinant(n)
    while true
        A = randmat(n)
        if abs(det(A)) < 0.1
            continue
        end
        A = A / abs(det(A))^1/n
        if det(A) > 0
            return A
        end
    end
end

35.5 μs

negative_determinant (generic function with 1 method)

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function negative_determinant(n)
    while true
        A = randmat(n)
        if abs(det(A)) < 0.1
            continue
        end
        A = A / abs(det(A))^1/n
        if det(A) < 0
            return A
        end
    end
end

35.7 μs

illustrate (generic function with 1 method)

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function illustrate(g, A, B)
    # function to plot side-by-side our image g transformed by A and B
    plot(
        heatmap(
            make_displayable(A * g), 
            legend=false, 
            color=:binary
        ),
        heatmap(
            make_displayable(B * g), 
            legend=false,
            color=:binary
        ),
        layout = (1, 2),
    )
end     

38.2 μs

The plots below show our starting image transformed by random matrices $A$ with positive and negative determinant.

Roughly speaking, if the matrix $A$ has positive determinant, so is orientation-preserving, then the transformed image "looks the same (but rotated)." This is shown on the left.

If the matrix $A$ has negative determinant, so is orientation-reversing, then the transformed image is somehow "the mirror image (but rotated)" of we started with. This is shown on the right.

5.8 μs

 
illustrate(g, positive_determinant(2), negative_determinant(2))

266 ms