Graphics Programs Reference
In-Depth Information
Indices of the elements of amatrix are displayedinthe sameorderas its dimensions:
the rownumber comes first, followedbythe columnnumber. Only one indexis needed
for the elements of avector.
Transpose
The transpose of amatrix
A
is denotedby
A
T
and definedas
A
i j
=
A
ji
The transpose operation thus interchanges the rows and columns of the matrix. If
applied to vectors, itturns a column vectorinto arowvector and
vice versa
.For
example, transposing
A
and
b
in Eq. (A9), we get
⎡
⎣
⎤
⎦
A
11
A
21
A
31
A
12
A
22
A
32
A
13
A
23
A
33
b
1
b
2
b
3
A
T
b
T
=
=
n
matrix issaid to be
symmetric
if
A
T
A
. This meansthat the elements
in the upper triangular portion (above the diagonalconnecting
A
11
and
A
nn
)ofa
symmetric matrix are mirroredinthe lower triangular portion.
An
n
×
=
Addition
The sum
C
=
A
+
B
of two
m
×
n
matrices
A
and
B
is definedas
C
i j
=
A
i j
+
B
i j
,
i
=
1
,
2
,...,
m
;
j
=
1
,
2
,...,
n
(A10)
Thus the elements of
C
areobtained by adding elements of
A
to the elements of
B
.
Note that additionis defined only formatrices thathave the same dimensions.
Multiplication
The scalar or
dot product c
=
a
·
b
of the vectors
a
and
b
,each of size
m
, is definedas
m
c
=
a
k
b
k
(A11)
k
=
1
It can also be writteninthe form
c
=
a
T
b
.
The matrix product
C
=
AB
of an
l
×
m
matrix
A
and an
m
×
n
matrix
B
is
definedby
m
C
i j
=
A
ik
B
kj
,
i
=
1
,
2
,...,
l
;
j
=
1
,
2
,...,
n
(A12)
k
=
1
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