Graphics Reference
In-Depth Information
In terms of vectors (as defined in Section 3.3),
c
ij
is the dot product of the
i
-th row
of
A
and the
j
-th column of
B
. Matrix multiplication is not commutative; that is, in
general,
AB
BA
. Division is not defined for matrices, but some square matrices
A
have an
inverse
, denoted inv(
A
)or
A
−
1
, with the property that
AA
−
1
=
A
−
1
A
=
=
I
.
Matrices that do not have an inverse are called
singular
(or
noninvertible
).
3.1.2
Algebraic Identities Involving Matrices
Given scalars
r
and
s
and matrices
A
,
B
, and
C
(of the appropriate sizes required
to perform the operations), the following identities hold for matrix addition, matrix
subtraction, and scalar multiplication:
A
+
B
=
B
+
A
A
+
(
B
+
C
)
=
(
A
+
B
)
+
C
A
−
B
=
A
+
(
−
B
)
−
−
=
(
A
)
A
s
(
A
±
B
)
=
s
A
±
s
B
(
r
±
s
)
A
=
r
A
±
s
A
r
(
s
A
)
=
s
(
r
A
)
=
(
rs
)
A
For matrix multiplication, the following identities hold:
AI
=
IA
=
A
A
(
BC
)
=
(
AB
)
C
A
(
B
±
C
)
=
AB
±
AC
(
A
±
B
)
C
=
AC
±
BC
(
s
A
)
B
=
s
(
AB
)
=
A
(
s
B
)
Finally, for matrix transpose the following identities hold:
B
)
T
A
T
B
T
(
A
±
=
±
(
s
A
)
T
s
A
T
=
(
AB
)
T
B
T
A
T
=
