Graphics Programs Reference
In-Depth Information
The definitionrequires the number of columns in
A
(the dimension
m
)tobeequalto
the number of rows in
B
. The matrix product can also be definedintermsofthedot
product.Representing the
i
th row of
A
as the vector
a
i
and the
j
th column of
B
as the
vector
b
j
, wehave
C
i j
=
a
i
·
b
j
(A13)
A square matrix of special importance is the identity or
unit matrix
⎡
⎣
⎤
⎦
1 00
···
0
0 1 0
···
0
001
0
. . .
.
.
.
.
0000 1
···
I
=
(A14)
It has the property
AI
=
IA
=
A
.
Inverse
n
matrix
A
, denotedby
A
−
1
, is defined to be an
n
The inverse of an
n
×
×
n
matrix
thathas the property
A
−
1
A
AA
−
1
=
=
I
(A15)
Determinant
|
|
The determinantofa square matrix
A
is a scalar denotedby
ordet(
A
). There is no
concise definition of the determinantforamatrix of arbitrary size.Westart with the
determinantofa2
A
×
2matrix, which is definedas
=
A
11
A
12
A
21
A
22
A
11
A
22
−
A
12
A
21
(A16)
The determinantofa3
×
3matrix is thendefinedas
−
+
A
11
A
12
A
13
A
21
A
22
A
23
A
31
A
32
A
33
A
22
A
23
A
32
A
33
A
21
A
23
A
31
A
33
A
21
A
22
A
31
A
32
=
A
11
A
12
A
13
Having established the pattern, wecan nowdefine the determinantofan
n
×
n
matrix
in terms of the determinantofan (
n
−
1)
×
(
n
−
1) matrix:
n
1)
k
+
1
A
1
k
M
1
k
|
A
| =
(
−
(A17)
k
=
1
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