Visualizing Matrices

1

• Visualizing Matrices
• John Peterson
• Yale University
• Languages for Mathematics
• Education
• www.haskell.org/edsl

2
Turning Math into Pictures

• I study languages that allow people to
• use computers in new ways.
• Interactive pictures give us a way of
• showing the underlying intuition behind
• the mathematics.

3
Review

• What you should know
• What a matrix is
• How to add and multiply matrices
• Determinates
• Inverting matrices
• Triangle Area theorem

4
Using a 2x2 Matrix Lens

• We use a matrix as a lens through which we
• view an image.
• A 2-d image is a mapping from points in the
  coordinate plane onto colors.

5
Adding a Matrix

• Now add a transformation between the observer
  (you) and the picture

x
Observation point
Image color
Image point
Transformation matrix
6
An Example

• How does the image change as parameters change?

2 4
1 2
x
Image point
Observation point
Image color
2 x Identity matrix
7
Live demo time
Run the demo 01-basic-matrix Make some
observations!
8
So what did we learn?

• Lines always remain straight
• Origin doesnt move 0 0 x M 0 0 no
  matter what M
• Angles may change
• Image may completely disappear
• But do we have any intuition about what
• a, b, c, and d do??

9
Whats in the Matrix

• One way to take apart the matrix is to apply it
• to some specific values, the x and y units,
• 1 0 and 0 1

x
???
10
Whats in the Matrix

• One way to take apart the matrix is to apply it
• to some specific values, the x and y units,
• 1 0 and 0 1

x
a b
11
Whats in the Matrix

• One way to take apart the matrix is to apply it
• to some specific values, the x and y units,
• 1 0 and 0 1

x
???
12
Whats in the Matrix

• One way to take apart the matrix is to apply it
• to some specific values, the x and y units,
• 1 0 and 0 1

x
c d
13
Units

• When we plot the point (a,b) on the
• warped picture were seeing the image of
• the x unit (c,d) is the image of the
• y unit.

Lets see how this works! Well mark the units in
the original picture and their images Run the
demo 02-show-points
14
Inverting a Matrix

• What does it mean to invert a matrix?
• Visually, it is how you undo an image warp.
• Lets try and invert the following matrix

1.31 1.14 0.26 -0.44
15
Inverting a Matrix
Transformed image
Original image
Transform this back to the original
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Inverting a Matrix

• Run 03-inverse and then check your answer

1.31 1.14 0.26 -0.44
?
x
17
Determinates

• What is the meaning of a determinate?
• Lets look at the image of the unit square.
• Note that the vertices of the square are
• (0,0) (a,b) (ac, bd) (b,d)
• Can you see the connection between
• the determinate and the square in
• the example 04-determinate?

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Determinates

• A (2x2) determinate indicates how the
• area of a figure changes under the image
• of a transformation.
• A determinate of 0 indicates that the
• figure collapses to a point or line.
• A 3x3 determinate indicates change in
• 3-dimensional volume.

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Measuring a Line

• If a 3x3 determinate measures the area of
• a triangle, we can use a 2x2 to measure
• the length of a line.
• Given a line from A to B, construct the
• following matrix

Run 05-length to see this
20
The Unit Circle

• The unit circle is a circle with radius 1
• centered at the origin.
• What is the diameter of this circle?
• Lets walk around the circle starting
• at (1,0). After walking a distance d, where
• are you?
• If this point is (a,b) then what is a b ?

2
2
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Sine and Cosine

• Run 06-sine.cs
• This shows you two functions
• if x is the distance traveled around the
• circle, then (cos(x), sin(x)) is the
• coordinate on the circle
• Note sin(x) and cos(x) are between
• -1 and 1

22
Factoring A 2x2 Matrix

• Some simple 2x2 matrices

X scaling
Y scaling
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Factoring A 2x2 Matrix

• Some simple 2x2 matrices

Skew
Rotation
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An Exercise

• Find the inverse and determinate of
• each one of these matrices.

M
M
s
x
M
M
r
y
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Decomposing a Matrix

• We can take apart a 2x2 matrix and
• express it as a product of these 4
• matrices

M x M x M x M
y
x
s
a
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Why Decompose?

• These simple matrices are easy to understand
  they have very useful
• effects on an image
• Run 07-factored to see how this works
• We can do anything to the image with
• the product of these 4 matrices that
• can be done with any other 2x2 matrix
• Run 08-factored-inverse to see more

27
Inverting a product matrix

• Lets invert our matrix
• This is right, isnt it?

-1
-1
-1
-1
-1
M x M x M x M
y
x
s
a
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Oops!

• We blew it! If a matrix is a product of
• other matrices, you have to reverse the
• order of the inverses

-1
-1
-1
-1
-1
M x M x M x M
x
y
s
a
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Translation

• Unfortunately a 2x2 matrix cant move
• things around - point (0,0) never moves.
• The solution use special 3x3 matrices

x y 1
Add a 1 to each point
Extra row
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Translation

Example multiply M x 5 2 1
T(2,3)
Run 09-translation to see this
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Matrix Computers

• Did you know that many computers have
• special hardware for multiplying matrices?
• Some computers do almost nothing except
• multiply lots and lots of these 3x3 (4x4 in
• most cases for 3D points)
• What do these computers look like?

32
Matrix Computers?
33
Matrix Computer!
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Building an Animation

• We can make a short animated movie
• by using matrices to move around
• images.
• Write equations like this for each of the
• parameters in the matrix
• x if time lt 2 then 0 else
• if time lt 4 then (time-2) 50 else
• 100

35
Some Math

• Its often easiest to map some interval
• of time onto the interval (0, 1).
• How would we map (2,5) onto (0,1)?
• (Hint you could solve this with matrices!)

f(t) (t-2)/3 f(2) 0, f(5) 1
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A Lame Animation
Run 11-animation to see this
sx if t lt 8 then 1 else (1-(t-8)/2) sy sx rot
if t lt 3 then 0 else if t lt 5 then
((t-3)/2)2pi else 0 skew if t lt 5 then
0 else sin((t-5)(t-5)) (xt,yt) if t lt 5 then
(30t, 10t) else (150, 50)
37
Thank You!

• We hope you enjoyed this presentation.