Biomedical Engineering Reference
In-Depth Information
2
3
M
11
M
12
:::
M
1c
M
21
M
22
:::
4
5
:
M
2c
M
¼
(A.1)
:
:
:
:
:
:
M
r1
:
:
:
:
M
rc
The typical element of the array,
M
ij
, is the
i
th element in the
j
th column; in this
text the elements
M
ij
will be real numbers or functions whose values are real
numbers. The transpose of the matrix
M
is denoted by
M
T
and is obtained from
M
by interchanging the rows and columns
2
3
M
11
M
21
:::
M
r1
4
5
:
M
12
M
22
:::
M
r2
M
T
¼
(A.2)
:
:
:
:
:
:
M
1c
:
:
:
:
M
rc
The operation of obtaining
M
T
from
M
is called transposition. In this text we are
interested in special cases of the
r
by
c
matrix
M
. These special cases are those of
the square matrix,
r
¼
c
¼
n
, the case of the row matrix,
r
¼
1,
c
¼
n
, and the case
of column matrix,
r
¼
n
,
c
¼
1. Further, the special sub-cases of interest are
n
¼
2,
n
1 reduces all three special cases to the trivial
situation of a single number or scalar. A square matrix
A
has the form
¼
3, and
n
¼
6; the sub-case
n
¼
2
4
3
5
;
A
11
A
12
:::
A
1n
A
21
A
22
:::
A
2n
A
¼
(A.3)
:
:
:
:
:
:
A
n1
:
:
:
:
A
nn
while row and column matrices,
r
and
c
, have the forms
2
4
3
5
c
1
c
2
:
:
c
n
r
¼½
r
1
r
2
:::
r
n
;
c
¼
;
(A.4)
respectively. The transpose of a column matrix is a row matrix, thus
c
T
¼½
c
1
c
2
:::
c
n
:
(A.5)
To save space in topics and papers the formof
c
in (A.5) is usedmore frequently than
the form in the second of (A.4). Wherever possible, square matrices will be denoted by
upper case boldface Latin letters, while row and column matrices will be denoted
by lower case boldface Latin letters as is the case in equations (A.3) and (A.4).
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