matrix math

Matrix Math

©Anthony Steed 1999
1


Overview
■

To revise

• Vectors
• Matrices
■

New stuff

• Homogenous co-ordinates
• 3D transformations as matrices

2


Vectors and Matrices
■

Matrix is an array of numbers with dimensions M (rows) by N (columns)

• 3 by 6 matrix
• element 2,3
is (3)

■

Vector can be considered a 1 x M matrix

 3 0 0 − 2 1 − 2
 1 1 3 4 1 −1 
−5 2 0 0 0 1 



•

v = ( x y z)

3


Types of Matrix
■

Identity matrices - I

1
0

■

0
1 

Diagonal

1
0
0
0


0
2
0
0

1
0
0
0

0
0
−1
0

0
1
0
0

■

0
0
1
0

0
0
0
− 4 

0
0
0
1 

Symmetric

a b c 
b d e 
c e f 


• Diagonal matrices
are (of course)
symmetric
• Identity matrices are
(of course) diagonal

4

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Operation on Matrices

■

Addition

• Done elementwise

a b   p q a + p b + q

+
=

c d   r s   c + r d + s

■

Transpose

• “Flip” (M by N becomes N by M)

T

1 4 9 1 5 6
 5 2 8 =  4 2 7
 6 7 3  9 7 3

 

5


Operations on Matrices
■

Multiplication

• Only possible to multiply of dimensions
– x1 by y1 and x2 by y2 iff y1 = x2
• resulting matrix is x1 by y2

– e.g. Matrix A is 2 by 3 and Matrix by 3 by 4
• resulting matrix is 2 by 4

– Just because A x B is possible doesn’t mean
B x A is possible!

6


Matrix Multiplication Order

■
■

A is n by k , B is k by m
C = A x B defined by

k

■

c = ∑a b

BxA
ijnot necessarily equal
il ljto AxB

l =1

*

 * * * * *  *

 *



 *
*



 *
 
=
 
 








.

 * * * * *  .

 .



 .
.


*
*
*
*

*


 . * 

 

=

 
 



.

 . . . . .  .
 * * * * * .



 .
.


*
*
*
*
*


 . .
 . *
=
 
 







7


Example Multiplications
2 1

 2 3 1 1 − 1 =  __ __ 
 − 1 0 1



 0 1   __ __ 


 2 − 2 3  0 0 1   __ __ __ 
 − 3 1 0  1 0 0  = __ __ __


 1 − 1 − 1 0 1 0  


  __ __ __ 
8


Inverse
■

If A x B = I and B x A = I then
A = B-1 and B = A-1

9


Anthony
AnthonySteed:
Steed:
EQ
EQneeds
needsfixing
fixing

3D Transforms

■

In 3-space vectors are transformed by 3 by 3 matrices

a b c 
( x y z )  d e f  = ( xa + yd + zg xb + ye + zf xc + yf + zi )
g h i 

■

Example?

10


Scale
■

■

Scale uses a diagonal matrix

 a 0 0
 0 b 0  = ( xa yb zc )
(
)
x
y
z
Scale by 2 along x and -2 along z
0 0 c



2 0 0 
( 3 4 5)  0 1 0  = ( 6 4 − 10)
 0 0 − 2
11


Rotation
■

Rotation about z axis

Y

 cos(θ ) sin(θ ) 0 
 - sin(θ ) cos(θ ) 0 


0
0 1
■ 

Note z values remain the same whilst x and y change

( cos( θ ) x - sin ( θ ) y

θ

sin ( θ ) x + cos( θ ) y )

( x y)

X

12


Rotation X, Y and Scale
■

About X

1 0
0 
 0 cos(θ ) - sin(θ ) 


0
sin(
θ
)
cos(
θ
)


■

About Y

■

Scale (should look familiar)

 a 0 0
 0 b 0
0 0 c



 cos(θ ) 0 - sin(θ ) 
 0 1
0 
 sin(θ ) 0 cos(θ ) 



13


Homogenous Points
■
■

Add 1D, but constrain that to be equal to 1 (x,y,z,1)
Homogeneity means that any point in 3-space can be represented by an
infinite variety of homogenous 4D points

• (2 3 4 1) = (4 6 8 2) = (3 4.5 6 1.5)
■

Why?

• 4D allows as to include 3D translation in matrix
form

14


Homogenous Vectors
■

Vectors != Points

■

Remember points can not be added

■

If A and B are points A-B is a vector

■

Vectors have form (x y z 0)

■

Addition makes sense

15

Translation in Homogenous Form

■

 1 0 0 0
 0 1 0 0
( x y z 1)  0 0 1 0  = ( x + a y + b z + c 1)
Note that the homogenous
is preserved (* * * 1), and aside from the translation
 a component

 b c 1
the matrix is I

16


Putting it Together

■
■

 R1 R 2
 R 4 R5
 R 7 R8
R is rotation and scale
components
T

1 T2

T is translation component

R3
R6
R9
T3

0
0
0

1

17


Order Matters
■

Composition order of transforms matters

• Remember that basic vectors change so
“direction” of translations changed

1

( X Y Z 1)  00
0


1

( X Y Z 1)  00
2


0
1
0
3

0
0
1
4

0
0
1
0

0
−1
0
0

0  1
0  0
0  0

1  0

0  1
0  0
0  0
1  2

0
0
1
0

0
−1
0
0

0
1
0
3

0
0
1
4

0
1 0


0 = ( X Z − Y 1)  0 1
0 0
0
2 3
1 


0
0
1
4

0
0  = ( X + 2 Z + 3 − Y + 4 1)
0
1 

0
1 0

0 = ( X + 2 Y + 4 Z + 4 1)  0 0
 0 −1
0
0 0
1 


0
−1
1

0

0
0  = ( X + 2 Z + 4 − Y − 4 1)
0
1 
18


Exercises
■

Show that rotation byΠ/2 about X and then Π/2 about Y is equivalent to Π/2 about Y then

Π/2 about X

■

Calculate the following matrix Π/2 about X then Π/2 about Y then Π/2 about Z (remember
“then” means multiply on the right). What is a simpler form of this matrix?

■

Compose the following matrix translate 2 along X, rotate Π/2 about Y, translate -2 along X.
Draw a figure with a few points (you will only need 2D) and then its translation under this
transformation.

19

Summary
■

Rotation, Scale, Translation

■

Composition of transforms

■

The homogenous form

20