Game Development Reference
In-Depth Information
2
4
1
3
5
T
2
3
2
3
1
4
7
10
4
5
6
4
5
=
2
5
8
11
,
(4.1)
7
8
9
3
6
9
12
10
11
12
Transposing matrices
2
3
5
T
2
3
a b c
d e f
g h i
a d g
b e h
c f i
4
4
5
=
.
(4.2)
For vectors, transposition turns row vectors into column vectors and
vice versa:
2
3
2
3
5
T
x
y
z
x
y
z
Transposing converts
between row and column
vectors
T
=
4
5
4
x y z
=
x y z
Transposition notation is often used to write column vectors inline in a
paragraph, such as [1,2,3]
T
.
Let's make two fairly obvious, but significant, observations regarding
matrix transposition.
(
M
T
)
T
=
M
, for a matrix
M
of any dimension. In other words, if we
transpose a matrix, and then transpose it again, we get the original
matrix. This rule also applies to vectors.
•
Any diagonal matrix
D
is equal to its transpose:
D
T
•
=
D
. This
includes the identity matrix
I
.
4.1.5 Multiplying a Matrix with a Scalar
A matrix
M
may be multiplied with a scalar k, resulting in a matrix of the
same dimension as
M
. We denote matrix multiplication with a scalar by
placing the scalar and the matrix side-by-side, usually with the scalar on
the left. No multiplication symbol is necessary. The multiplication takes
place in a straightforward fashion: each element in the resulting matrix k
M
is the product of k and the corresponding element in
M
. For example,
2
4
m
11
m
12
m
13
3
5
2
4
km
11
km
12
km
13
3
5
Multiplying a 4 × 3
matrix by a scalar
m
21
m
22
m
23
m
31
m
32
m
33
m
41
m
42
m
43
km
21
km
22
km
23
km
31
km
32
km
33
km
41
km
42
km
43
k
M
= k
=
.
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