4 1 Polynomial Matrices
(2) For arbitrary polynomials f (λ), d(λ), there exists a pair of polynomials q(λ),
r(λ) such that
f (λ) = q(λ)d(λ) +r (λ), (1.3)
where
degr(λ) < deg d(λ). (1.4)
Here the polynomial q(λ) is called the entire part, and the polynomial r(λ) the
remainder at the division of f (λ) by d(λ).
(3) If r (λ) = 0, i.e.,
f (λ) = q(λ)d(λ) (1.5)
we say that the polynomial q(λ) is a divisor of the polynomial f (λ).
Let the polynomials f
1
(λ), f
2
(λ) ∈ F[λ] be given. Then the polynomial p(λ) is
called a common divisor (CD) of f
1
(λ) and f
2
(λ) if it is a divisor of each of them.
A CD of greatest possible degree is called a greatest common divisor (GCD).
The GCD of the polynomials f
1
(λ) and f
2
(λ) can be represented in the form
p(λ) = f
1
(λ)m
1
(λ) + f
2
(λ)m
2
(λ), (1.6)
where m
1
(λ), m
2
(λ) are certain polynomials.
The monic GCD of the polynomials f
1
(λ) and f
2
(λ) is uniquely defined.
(4) The polynomials f
1
(λ) and f
2
(λ) are called coprime if they do not possess
common roots. The monic GCD of coprime polynomials is equal to one. For the
fact that the polynomials f
1
(λ) and f
2
(λ) are coprime, the following condition is
necessary and sufficient: There exist polynomials m
1
(λ) and m
2
(λ) such that
f
1
(λ)m
1
(λ) + f
2
(λ)m
2
(λ) = 1. (1.7)
If
f
1
(λ) = p(λ)
˜
f
1
(λ), f
2
(λ) = p(λ)
˜
f
2
(λ), (1.8)
where p(λ) is the GCD, then the polynomials
˜
f
1
(λ) and
˜
f
2
(λ) are coprime.
1.2 Polynomial Matrices
(1) The set of constant n × m matrices with entries from C, R, F are denoted by
C
nm
, R
nm
, F
nm
, respectively. The set of n × m matrices with entries from C[λ],
R[λ], F[λ] are denoted by C
nm
[λ], R
nm
[λ], F
nm
[λ], respectively. In the following,
these matrices are called polynomial matrices. A polynomial matrix A(λ) ∈ F
nm
[λ]
can be written in the form