Biomedical Engineering Reference
In-Depth Information
column matrices is n-tuples. This phrase will be used here because it is descriptive
and inclusive. A zero n-tuple is an n-tuple whose entries are all zero; it is denoted by
0
¼
[0, 0,
...
, 0]. The multiplication of an n-tuple
r
by a scalar
a
is defined as the
multiplication of every element of the n-tuple
r
by the scalar
a
, thus
a
r
¼
[
a
r
1
,
a
r
2
,
...
a
r
n
]. As with square matrices, it is then easy to show for n-tuples that 1
r
¼
r
,
,
0
. The addition of n-tuples is only defined for n-
tuples with the same n. The sum of two n-tuples,
r
and
t
, is denoted by
r
1
r
¼
r
,0
r
¼
0
, and
a
0
¼
þ
t
, where
r
þ
t ¼ t þ r
, and associative,
r þ
(
t þ u
)
¼
(
r þ t
)
þ u
. The following distributive
laws connect n-tuple addition and n-tuple multiplication by scalars, thus
a
(
r þ t
)
¼ ar þ at
and (
þ
t
¼
[
r
1
þ
t
1
,
r
2
þ
t
2
,
...
,
r
n
þ
t
n
]. Row-matrix addition is commutative,
r
are scalars. Negative n-tuples
may be created by employing the definition of n-tuple multiplication by a scalar,
a
a þ b
)
r ¼ ar þ br
, where
a
and
b
r
¼
[
a
r
1
,
a
r
2
,
...
,
a
r
n
], in the special case when
a ¼
1. In this case the definition
of addition of n-tuples,
r
þ
t
¼
[
r
1
þ
t
1
,
r
2
þ
t
2
,
...
,
r
n
þ
t
n
], can be extended to
include the subtraction of n-tuples
r
t
.
Two n-tuples may be employed to create a square matrix. The square matrix
formed from
r
and
t
is called the
open
product of the n-tuples
r
and
t
; it is denoted
by
r
t
, and the difference between n-tuples,
r
t
, and defined by
2
4
3
5
:
r
1
t
1
r
1
t
2
...
r
1
t
n
r
2
t
1
r
2
t
2
...
r
2
t
n
r
t
¼
(A.30)
:
:
...
:
r
n
t
1
:
...
r
n
t
n
The American physicist J. Willard Gibbs introduced the concept of the open
product of vectors calling the product a
dyad
. This terminology is still used in some
topics and the notation is spoken of as the
dyadic
notation. The trace of this square
matrix, tr{
r
t
} is the scalar product of
r
and
t
,
tr
f
r
t
g¼
r
t
¼
r
1
t
1
þ
r
2
t
2
þþ
r
n
t
n
:
(A.31)
In the special case of
n
¼
3, the skew-symmetric part of the open product
r
t
,
2
3
0
r
1
t
2
r
2
t
1
r
1
t
3
r
3
t
1
1
2
4
5
;
r
2
t
1
r
1
t
2
0
r
2
t
3
r
3
t
2
(A.32)
r
3
t
1
r
1
t
3
r
3
t
2
r
2
t
3
0
provides the components of the cross product of
r
and
t
, denoted by
r
t
, and
r
2
t
1
]
T
. These points concerning the
written as
r
t
¼
[
r
2
t
3
r
3
t
2
,
r
3
t
1
r
1
t
3
,
r
1
t
2
dot product
r
t
and cross product
r
t
will be revisited later in this Appendix.
Example A.4.1
Given the n-tuples
a ¼
[1, 2, 3] and
b ¼
[4, 5, 6], construct the open product
matrix,
, the skew-symmetric part of the open product matrix, and trace of the
open product matrix.
a
b
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