In analyzing a left or right matrix-fraction model of a system, the Smith form can be used to determine whether, or not, the numerator and denominator matrices are coprime. If the matrices concerned are coprime, then there is no lower-order matrix-fraction model that will generate the same corresponding transfer-function matrix representation. Two polynomial matrices X(s) and Y(s), of dimensions r × r and r × m, respectively, are said to be coprime, or relatively left prime if the Smith form of the combined r × (r+m) matrix [X(s) Y(s)] is equal to the matrix [Ir 0]. If this is not true, then X(s) and Y(s) have a left matrix greatest common divisor (GCD) L(s), where the determinant of L(s) is a nonzero polynomial in the variable s. This means that X(s) and Y(s) can be expressed as
where (s) and (s) are polynomial matrices. A representation of a rational polynomial matrix G(s) as the left matrix fraction G(s)=DL(s)-1NL(s), where DL(s) and NL(s) are left coprime, is known as a left coprime factorization (LCF).
Consider the transfer-function matrix
which can be expressed as the left matrix-fraction model
where the degree of the determinant of the denominator matrix DL(s) is 4. Since both DL(s) and NL(s) can equally be expressed as
then G(s) can be realized as the lower-order left matrix fraction
since now the degree of the determinant of is only 3. Also, the determinant of the left matrix greatest common divisor L(s) is (s+3), which is a common factor that would cancel out when forming G(s) from the original left matrix-fraction representation DL(s)-1NL(s). Similar arguments can be given with respect to two polynomial matrices X(s) and V(s), of dimensions r × r and l × r, respectively, which are said to be coprime, or relatively right prime, if the Smith form of the combined (r+l) × r matrix is equal to the matrix . If this is not true, then X(s) and V(s) have a right matrix greatest common divisor R(s), where the determinant of R(s) is a nonzero polynomial in the variable s. This means that X(s) and V(s) can be expressed as
where (s) and (s) are polynomial matrices. A representation of a rational polynomial matrix G(s) as the right matrix fraction G(s)=NR(s) DR(s)-1, where DR(s) and NR(s) are right coprime, is known as a right coprime factorization (RCF).
(s) and 
(s) are polynomial matrices.
A representation of a rational polynomial matrix G(s) as the left matrix fraction G(s)=DL(s)-1NL(s), where DL(s) and NL(s) are left coprime, is known as a left coprime factorization (LCF).
is only 3. Also, the determinant of the left matrix greatest common divisor L(s) is (s+3), which is a common factor that would cancel out when forming G(s) from the original left matrix-fraction representation DL(s)-1NL(s).
Similar arguments can be given with respect to two polynomial matrices X(s) and V(s), of dimensions r × r and l × r, respectively, which are said to be coprime, or relatively right prime, if the Smith form of the combined (r+l) × r matrix 
is equal to the matrix 
. If this is not true, then X(s) and V(s) have a right matrix greatest common divisor R(s), where the determinant of R(s) is a nonzero polynomial in the variable s. This means that X(s) and V(s) can be expressed as
(s) and 
(s) are polynomial matrices.
A representation of a rational polynomial matrix G(s) as the right matrix fraction G(s)=NR(s) DR(s)-1, where DR(s) and NR(s) are right coprime, is known as a right coprime factorization (RCF).