Game Development Reference

In-Depth Information

FF

F

11

21

31

F

FF

12

22

32

T

.

(3.2)

F

=

F

FF

13

23

33

F

FF

14

24

34

matrices), scalar multiplica-

tion is defined for matrices. Given a scalar
a
and an
nm

As with vectors (which can be thought of as

n

×

1

×

matrix
M
, the product

a
M
is given by

aM

aM

aM

11

12

1

m

aM

aM

aM

21

22

2

m

.

(3.3)

a

MM

=

a

aM

aM

aM

n

1

n

2

nm

Also in a manner similar to vectors, matrices add entrywise. Given two
nm

×

ma-

trices
F
and
G
, the sum

+

G
is given by

F

+

GFG

+

FG

+

F

11

11

12

12

1

m

1

m

FG FG

+

+

F G

+

21

21

22

22

2

m

2

m

.

(3.4)

F

+=

G

FG F G

+

+

F G

+

n

1

n

1

n

2

n

2

nm

nm

Two matrices
F
and
G
can be multiplied together, provided that the number

of columns in
F
is equal to the number of rows in
G
. If
F
is an
nm

×

matrix and

()

G
is an
mp

×

matrix, then the product
FG
is an
np

×

matrix whose

,
ij
entry is

given by

m

(

)

FG

=

F G

.

(3.5)

ij

ik

kj

k

=

1

()

Another way of looking at this is that the

,
ij
entry of
FG
is equal to the dot

product of the
i
-th row of
F
and the
j
-th column of
G
.

There is an
nn

matrix called the
identity
matrix, denoted by
I
, for which

×

MIIMM
for any
nn

==

matrix
M
. The identity matrix has the form

×

n

n

10

0

01

0

I

=

.

(3.6)

n

00

1

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