Graphics Reference
In-Depth Information
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We will discover in later chapters that a 4
4 matrix is the largest matrix we will
require to represent a 3D rotation. Now let's look at some of the ways we manipulate
matrices.
4.3 The Transpose of a Matrix
One useful matrix operation is the
transpose
where every element
a
row,col
is ex-
changed with its transpose
a
col,row
, and is written
A
T
T
=[
a
col,row
]
.
For example, here is a matrix
A
and its transpose
A
T
=[
a
row,col
]
12
34
,
13
24
.
A
T
A
=
=
A
T
. Such a matrix is called a
It is possible that the elements of
A
are such that
A
=
symmetric
matrix, and we will examine this later.
4.4 The Identity Matrix
As mentioned above, the
identity matrix
I
is a matrix such that
IA
A
.
The three identity matrices we will encounter in later chapters are
=
AI
=
⎡
⎣
⎤
⎦
⎡
⎤
1000
0100
0010
0001
10
01
,
100
010
001
⎣
⎦
,
and it should be obvious that
I
T
=
I
.
4.5 Adding and Subtracting Matrices
It is possible to add and subtract matrices so long as they have the same number of
rows and columns. For example, in matrix notation
A
±
B
=[
a
row,col
±
b
row,col
]
.
For example:
56
78
,
12
34
A
=
B
=
then
68
10
,
44
44
.
+
=
−
=
A
B
A
B
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