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components. This also intuitively suggests that the subdeterminants of
A
with
size 2 are principal components of a corresponding matrix from the viewpoint
of statistical dependence.
Recall that statistical independence of two attributes is equivalent to a
corresponding matrix with rank being 1. A matrix with rank being 2 gives a
context-dependent independence, which means three values of two attributes
are independent, but two values of two attributes are dependent.
3.3 Elementary Divisors and Elementary Transformation
Let us define the following three elementary (row/column)transformations of
a corresponding matrix:
1. Exchange two rows (columns),
i
0
and
j
0
(
P
(
i
0
,j
0
)).
2. Multiply
1)).
3. Multiply
t
to a row (column)
j
0
(
i
0
)andaddittoarow
i
0
(
j
0
).
(
W
(
i
0
,j
0
,t
)).
Then, three transformations have several interesting characteristics.
−
1 to a row (column)
i
0
(
T
(
i
0
;
−
Proposition 3.
Matrices corresponding to three elementary transformations
are regular.
Proposition 4.
Three elementary transformations do not change the rank of
a corresponding matrix.
A denote a matrix transformed by finite steps of three
Proposition 5.
Let
operations. Then,
rank A
=
rankA, d
r
(
A
)=
d
r
(
A
)
,
where
r
denotes the rank of matrix
A
.
Then, from the results of linear algebra, the following interesting result is
obtained.
Theorem 4.
With the finite steps of elementary transformations, a given cor-
responding matrix is transformed into
⎛
⎞
e
1
⎝
⎠
e
2
.
.
.
O
e
r
O O
A
=
,
d
j
(
A
)
d
j−
1
(
A
)
(
d
0
(
A
)=1)
and r denotes the rank of a corresponding
matrix. Then, the determinant is decomposed into the product of e
j
.
where e
j
=
d
r
(
A
)=
d
r
(
A
)=
e
1
e
2
···e
r
.
