4. Co-prime MFD
•
irreducible?
•
for an si-so system, no common root
between the numerator and the denominator
Ex.
[
]
[
]
.
),
1
(
5
)
2
(
)
2
(
)
1
(
5
)
2
(
)
1
(
5
)
(
1
2
1
2
2
e
irreducibl
s
s
s
s
s
s
s
G
+
⋅
+
=
+
⋅
+
=
+
+
=
−
−
Characterization of the common factors (divisors)?
Any common factor (divisor) between the numerator
and the denominator must be a non-zero constant.
Generalization?

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i. Unimodular Matrix
Ex.
{
}
,
1
1
0
1
1
1
0
1
1
0
1
1
)
(
)
(
)
(
1
0
1
)
(
1
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
=
⎥
⎦
⎤
⎢
⎣
⎡
−
=
⎥
⎦
⎤
⎢
⎣
⎡
−
=
=
⎥
⎦
⎤
⎢
⎣
⎡
=
−
s
s
s
s
s
s
s
L
s
L
adj
s
L
s
s
L
T
Definition
.
Let
L
(
s
) be a non-singular square matrix
of polynomials. The matrix
L
(
s
) is said to be
unimodular if its inverse is a matrix of polynomials.
not unimodular.

Any non-singular square matrix of constants is unimodular.
Any others?
Ex.
{
}
,
1
0
)
3
(
1
1
)
3
(
0
1
1
1
)
(
)
(
)
(
1
0
)
3
(
1
)
(
2
2
1
2
⎥
⎦
⎤
⎢
⎣
⎡
−
−
=
⎥
⎦
⎤
⎢
⎣
⎡
−
−
=
=
⎥
⎦
⎤
⎢
⎣
⎡
−
=
−
s
s
s
L
s
L
adj
s
L
s
s
L
T
unimodular.
Theorem
. A non-singular square matrix of polynomials
is unimodular if and only if its determinant is a non-
zero constant.

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