Information Technology Reference
In-Depth Information
⎧
⎨
⎫
⎬
de f g
a
12
df i
a
11 12
⊥
3
⊥
=
,
.
b
4
⊥
5
⊥
c
⊥
14 13
⎩
⎭
c
67
⊥
8
h
15
⊥⊥
Finally, we can define “
inflating operation
” that is defined for index sets
⊂
⊂
I
⊂
⊂
I
K
P
and
L
Q
by
(
P
,
Q
)
A
=
(
P
,
Q
)
[
K
,
L
,
{
a
k
i
,
l
j
}] = [
P
,
Q
,
{
b
p
r
,
q
s
}]
,
where
a
k
i
,
l
j
,
if
p
r
=
k
i
∈
K
and
q
s
=
l
j
∈
L
b
p
r
,
q
s
=
⊥
,
otherwise
3.6 EIMs, Determinants and Permanents
As it is well-known, to each standard matrix can be juxtaposed a number, called
determinant. Also, it is known that if we change the places of two rows or two
columns of a standard matrix, the determinant of the new matrix will coincide with
the former, but with an opposite sign (i.e., sign “+” is changed with sign “
”orvice
versa). The same change of two rows or two columns of an IM, however, does not
change the form of the new IM. For example, the two IMs from the above example
satisfy the equality
−
di f
a
11
df i
a
11 12
⊥
12
⊥
=
.
c
⊥
13 14
c
⊥
14 13
h
15
⊥⊥
h
15
⊥⊥
Of course, we can juxtapose a determinant only to a matrix with elements being
real (complex) numbers. Having in mind the above equality, we can conclude that
for IMs there is no possibility to juxtapose a determinant. On the other hand, to each
IM with elements being real (or complex) numbers, we can juxtapose a permanent
(see e.g., [42]).
Now, we extend this possibility to each EIM. For this reason, we must define some
evaluating function
Φ
:
X
→
R
, such that it is an identity in the case
X
=
R
.
Let
l
1
...
l
j
...
l
n
k
1
a
k
1
,
l
1
...
a
k
1
,
l
j
...
a
k
1
,
l
n
.
.
.
.
.
...
. . .
A
=[
K
,
L
,
{
a
k
i
,
l
j
}] =
,
k
i
a
k
i
,
l
1
a
k
i
,
l
j
...
a
k
i
,
l
n
.
.
.
.
.
...
. . .
k
m
a
k
m
,
l
1
...
a
k
m
,
l
j
...
a
k
m
,
l
n
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