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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