1. Introduction
Consider a linear time invariant system
to be identified with the pair of matrices where , and or the fields of the real or complex numbers. If state-feedback is applied to system (▭), Rosenbrock's Theorem on pole assignment (see [1]) characterizes for the closed-loop system
the invariant factors of its state-space matrix . This result can be seen as the solution of an inverse problem; that of finding a non-singular polynomial matrix with prescribed invariant factors and left Wiener–Hopf factorization indices at infinity. To see this we recall that the invariant factors form a complete system of invariants for the finite equivalence of polynomial matrices (this equivalence relation will be revisited in Section ▭) and it will be seen in Section ▭ that any polynomial matrix is left Wiener–Hopf equivalent at infinity to a diagonal matrix , where the non-negative integers (that can be assumed in non-increasing order) form a complete system of invariants for the left Wiener–Hopf equivalence at infinity. Consider now the transfer function matrix of (▭). This is a rational matrix that can be written as an irreducible matrix fraction description , where and are right coprime polynomial matrices. In the terminology of [2], is a polynomial matrix representation of (▭), concept that is closely related to that of polynomial model introduced by Fuhrmann (see for example [3] and the references therein). It turns out that all polynomial matrix representations of a system are right equivalent (see [2], [3]), that is, if and are polynomial matrix representations of the same system there exists a unimodular matrix such that . Therefore all polynomial matrix representations of (▭) have the same invariant factors, which are the invariant factors of except for some trivial ones. Furthermore, all polynomial matrix representations also have the same left Wiener– Hopf factorization indices at infinity, which are equal to the controllability indices of (▭) and (▭), because the controllability indices are invariant under feedback. With all this in mind it is not hard to see that Rosenbrock's Theorem on pole assignment is equivalent to finding necessary and sufficient conditions for the existence of a non-singular polynomial matrix with prescribed invariant factors and left Wiener–Hopf factorization indices at infinity. This result will be precisely stated in Section ▭ once all the elements that appear are properly defined. In addition, there is a similar result to Rosenbrock's Theorem on pole assignment but involving the infinite structure (see [4]).
Our goal is to generalize both results (the finite and infinite versions of Rosenbrock's Theorem) for rational matrices defined on arbitrary fields via local rings. This will be done in Section ▭ and an extension to arbitrary fields of the concept of Wiener–Hopf equivalence will be needed. This concept is very well established for complex valued rational matrix functions (see for example [5], [6]). Originally it requires a closed contour, , that divides the extended complex plane () into two parts: the inner domain () and the region outside (), which contains the point at infinity. Then two non-singular complex rational matrices and , with no poles and no zeros in , are said to be left Wiener–Hopf equivalent with respect to if there are matrices and with no poles and no zeros in and , respectively, such that
It can be seen, then, that any non-singular complex rational matrix is left Wiener–Hopf equivalent with respect to to a diagonal matrix
where is any complex number in and are integers uniquely determined by . They are called the left Wiener–Hopf factorization indices of with respect to (see again [5], [6]). The generalization to arbitrary fields relies on the following idea: We can identify and with two sets and , respectively, of maximal ideals of . In fact, to each we associate the ideal generated by , which is a maximal ideal of . Notice that is also a prime polynomial of but and , as defined, cannot contain the zero ideal, which is prime. Thus we are led to consider the set of maximal ideals of . By using this identification we define the left Wiener–Hopf equivalence of rational matrices over an arbitrary field with respect to a subset of , the set of all maximal ideals of . In this study local rings play a fundamental role. They will be introduced in Section ▭. Localization techniques have been used previously in the algebraic theory of linear systems (see, for example, [7]). In Section ▭ the algebraic structure of the rings of proper rational functions with prescribed finite poles is studied (i.e., for a fixed the ring of proper rational functions with for all ). It will be shown that if there is an ideal generated by a linear polynomial outside then the set of proper rational functions with no poles in is an Euclidean domain and all rational matrices can be classified according to their Smith–McMillan invariants. In this case, two types of invariants live together for any non-singular rational matrix and any set : its Smith–McMillan and left Wiener–Hopf invariants. In Section ▭ we show that a Rosenbrock-like Theorem holds true that completely characterizes the relationship between these two types of invariants.
2. Preliminaries
In the sequel will denote the ring of polynomials with coefficients in an arbitrary field and the set of all maximal ideals of , that is,
Let be a monic irreducible non-constant polynomial. Let be the multiplicative subset of whose elements are coprime with . We denote by the quotient ring of by ; i.e., :
This is the localization of at (see [8]). The units of are the rational functions such that and . Consequentially,
For any , let
This is a ring whose units are the rational functions such that for all ideals , and . Notice that, in particular, if then and if then , the field of rational functions.
Moreover, if is a non-constant polynomial whose prime factorization, , satisfies the condition that for all , we will say that factorizes in or has all its zeros in . We will consider that the only polynomials that factorize in are the constants. We say that a non-zero rational function factorizes in if both its numerator and denominator factorize in . In this case we will say that the rational function has all its zeros and poles in . Similarly, we will say that has no poles in if and for all ideals . And it has no zeros in if for all ideals . In other words, it is equivalent that has no poles and no zeros in and that is a unit of . So, a non-zero rational function factorizes in if and only if it is a unit in .
Let denote the set of matrices with elements in . A matrix is invertible in if all its elements are in and its determinant is a unit in . We denote by the group of units of .
Let . Notice that
1. If then and .
2. and .
For any the ring is a principal ideal domain (see [9]) and its field of fractions is . Two matrices are equivalent with respect to if there exist matrices such that . Since is a principal ideal domain, for all non-singular (see [10]) there exist matrices such that
with (“” stands for divisibility) monic polynomials factorizing in , unique up to multiplication by units of . The diagonal matrix is the Smith normal form of with respect to and are called the invariant factors of with respect to . Now we introduce the Smith–McMillan form with respect to . Assume that is a non-singular rational matrix. Then with and monic, factorizing in . Let be the Smith normal form with respect to of , i.e., invertible in and monic polynomials factorizing in . Then
where are irreducible rational functions, which are the result of dividing by and canceling the common factors. They satisfy that , are monic polynomials factorizing in . The diagonal matrix in (▭) is the Smith–McMillan form with respect to . The rational functions , , are called the invariant rational functions of with respect to and constitute a complete system of invariants of the equivalence with respect to for rational matrices.
In particular, if then , the matrices are unimodular matrices, (▭) is the global Smith–McMillan form of a rational matrix (see [11] or [1] when or ) and are the global invariant rational functions of .
From now on rational matrices will be assumed to be non-singular unless the opposite is specified. Given any we say that an non-singular rational matrix has no zeros and no poles in if its global invariant rational functions are units of . If its global invariant rational functions factorize in , the matrix has its global finite structure localized in and we say that the matrix has all zeros and poles in . The former means that and the latter that because and are non-zero constants. The following result clarifies the relationship between the global finite structure of any rational matrix and its local structure with respect to any .
There exist such that the global invariant rational functions of are , and .
There exist matrices invertible in such that
uid15i.e., are the invariant rational functions of with respect to .
and with units of , for .
2 3. There exist unimodular matrices such that
with irreducible rational functions such that and are monic polynomials. Write such that factorize in and factorize in . Then
with and invertible in . Since the Smith–McMillan form with respect to is unique we get that .
3 1. Write (▭) as
It follows that with and .
1.08
Conversely, if and are equivalent with respect to and with respect to then, by the necessity of this result, they are equivalent with respect to , with respect to and with respect to . Let be the invariant rational functions of and with respect to , be the invariant rational functions of and with respect to and be the invariant rational functions of and with respect to . By Corollary ▭ must be the invariant rational functions of and with respect to . Therefore, and are equivalent with respect to .
Let be the ring of proper rational functions, that is, rational functions with the degree of the numerator at most the degree of the denominator. The units in this ring are the rational functions whose numerators and denominators have the same degree. They are called biproper rational functions. A matrix is said to be biproper if it is a unit in or, what is the same, if its determinant is a biproper rational function.
Recall that a rational function has a pole (zero) at if has a pole (zero) at 0. Following this idea, we can define the local ring at as the set of rational functions, , such that does not have 0 as a pole, that is, . If with , , where stands for “degree of”, then
As , then
Thus, this set is the ring of proper rational functions, .
Two rational matrices are equivalent at infinity if there exist biproper matrices such that . Given a non-singular rational matrix (see [11]) there always exist such that
where are integers. They are called the invariant orders of at infinity and the rational functions are called the invariant rational functions of at infinity.
1.05
3. Structure of the ring of proper rational functions with prescribed finite poles
Let . Any non-zero rational function can be uniquely written as where is an irreducible rational function factorizing in and is a unit of . Define the following function over (see [11], [12]):
This mapping is not a discrete valuation of if : Given two non-zero elements it is clear that ; but it may not satisfy that . For example, let . Put and . We have that , but .
However, if and where , , the map
defined via if and if is a discrete valuation of .
Consider the subset of , , consisting of all proper rational functions with poles in , that is, the elements of are proper rational functions whose denominators are coprime with all the polynomials such that . Notice that if and only if where:
is a polynomial factorizing in ,
is an irreducible rational function and a unit of ,
or equivalently .
In particular implies that . The units in are biproper rational functions , that is , with factorizing in . Furthermore, is an integral domain whose field of fractions is provided that (see, for example, Prop.5.22[11]). Notice that for , .
Assume that there are ideals in generated by linear polynomials and let be any of them. The elements of can be written as where factorizes in , is a unit in and . If is algebraically closed, for example , and the previous condition is always fulfilled.
The divisibility in is characterized in the following lemma.
Conversely, if then there is , factorizing in , such that . Write where is an irreducible fraction representation of , i.e., after canceling possible common factors. Thus and
Then and .
Notice that condition (▭) means that in and condition (▭) means that in . So, in if and only if simultaneously in and .
It follows from this Lemma that if are coprime in both rings and then are coprime in . The following example shows that the converse is not true in general.
Now, we have the following property when there are ideals in , , generated by linear polynomials.
Now suppose that and . Let . We have that . Thus is not a unit in and divides and because and . This is again impossible and either or .
The above lemmas yield a characterization of coprimeness of elements in when excludes at least one ideal generated by a linear polynomial.
Following the same steps as in p. 11[12] and p. 271[11] we get the following result.
The following examples show that if all ideals generated by polynomials of degree one are in , the ring may not be a Bezout domain. Thus, it may not be a Euclidean domain. Even more, it may not be a greatest common divisor domain.
.
If and then or with a non-zero constant. Then and if and only if and or and . So, the list of common divisors of and is:
If there would be a greatest common divisor, say , then because must be a multiple of and . Thus such a greatest common divisor should be either or , but does not divide neither of them because
Thus, and do not have greatest common divisor.
3.1. Smith–McMillan form
A matrix is invertible in if and its determinant is a unit in both rings, and , i.e., if and only if .
Two matrices are equivalent in if there exist invertible in such that
If there are ideals in generated by linear polynomials then is an Euclidean ring and any matrix with elements in admits a Smith normal form (see [10], [11] or [12]). Bearing in mind the characterization of divisibility in given in Lemma ▭ we have
with monic polynomials factorizing in and integers such that .
Under the hypothesis of the last theorem form a complete system of invariants for the equivalence in and are called the invariant rational functions of in . Notice that because divides .
Recall that the field of fractions of is when . Thus we can talk about equivalence of matrix rational functions. Two rational matrices are equivalent in if there are invertible in such that
When all ideals generated by linear polynomials are not in , each rational matrix admits a reduction to Smith–McMillan form with respect to .
with coprime for all such that , are monic polynomials factorizing in , divides for while divides for .
The elements of the diagonal matrix, satisfying the conditions of the previous theorem, constitute a complete system of invariant for the equivalence in of rational matrices. However, this system of invariants is not minimal. A smaller one can be obtained by substituting each pair of positive integers by its difference .
Suppose that , are monic and coprime polynomials factorizing in such that and . And suppose also that are integers such that . If for all , we define non-negative integers and for . If for all , we define and . Otherwise there is an index such that
Define now the non-negative integers as follows:
Notice that . Moreover,
In any case and are elements of . Now, on the one hand are coprime and or . This means (Lemma ▭) that are coprime for all . On the other hand and . Then (Lemma ▭) divides . Similarly, since and , it follows that divides .
We call , , the invariant rational functions of in .
There is a particular case worth considering: If then and . In this case, we obtain the invariant rational functions of at infinity (recall (▭)).
4. Wiener–Hopf equivalence
The left Wiener–Hopf equivalence of rational matrices with respect to a closed contour in the complex plane has been extensively studied ([5] or [6]). Now we present the generalization to arbitrary fields ([13]).
This is, in fact, an equivalence relation as it is easily seen. It would be an equivalence relation even if no condition about the union and intersection of and were imposed. It will be seen later on that these conditions are natural assumptions for the existence of unique diagonal representatives in each class.
The right Wiener–Hopf equivalence with respect to is defined in a similar manner: There are invertible matrices in and in such that
In the following only the left Wiener–Hopf equivalence will be considered, but, by transposition, all results hold for the right Wiener–Hopf equivalence as well.
The aim of this section is to obtain a complete system of invariants for the Wiener–Hopf equivalence with respect to of rational matrices, and to obtain, if possible, a canonical form.
There is a particular case that is worth-considering: If and , the invertible matrices in are the biproper matrices and the invertible matrices in are the unimodular matrices. In this case, the left Wiener–Hopf equivalence with respect to is the so-called left Wiener–Hopf equivalence at infinity (see [14]). It is known that any non-singular rational matrix is left Wiener–Hopf equivalent at infinity to a diagonal matrix where are integers, that is, for any non-singular there exist both a biproper matrix and a unimodular matrix such that
where are integers uniquely determined by . They are called the left Wiener–Hopf factorization indices at infinity and form a complete system of invariants for the left Wiener–Hopf equivalence at infinity. These are the basic objects that will produce the complete system of invariants for the left Wiener–Hopf equivalence with respect to .
For polynomial matrices, their left Wiener–Hopf factorization indices at infinity are the column degrees of any right equivalent (by a unimodular matrix) column proper matrix. Namely, a polynomial matrix is column proper if it can be written as with non-singular, non-negative integers and a polynomial matrix such that the degree of the th column of smaller than , . Let be non-singular polynomial. There exists a unimodular matrix such that is column proper. The column degrees of are uniquely determined by , although is not (see [14], p. 388[15], [16]). Since is column proper, it can be written as with non-singular, and the degree of the th column of smaller than , . Then . Put . Since is non-singular and is a strictly proper matrix, is biproper, and where .
The left Wiener–Hopf factorization indices at infinity can be used to associate a sequence of integers with every non-singular rational matrix and every . This is done as follows: If then it can always be written as such that the global invariant rational functions of factorize in and or, equivalently, the global invariant rational functions of factorize in (see Proposition ▭). There may be many factorizations of this type, but it turns out (see Proposition 3.2[4] for the polynomial case) that the left factors in all of them are right equivalent. This means that if with the global invariant rational functions of and factorizing in and the global invariant rational functions of and factorizing in then there is a unimodular matrix such that . In particular, and have the same left Wiener–Hopf factorization indices at infinity. Thus the following definition makes sense:
,
the global invariant rational functions of factorize in , and
the global invariant rational functions of factorize in .
Then the left Wiener–Hopf factorization indices of with respect to are defined to be the left Wiener–Hopf factorization indices of at infinity.
In the particular case that , we can put and . Therefore, the left Wiener–Hopf factorization indices of with respect to are the left Wiener–Hopf factorization indices of at infinity.
We prove now that the left Wiener–Hopf equivalence with respect to can be characterized through the left Wiener–Hopf factorization indices with respect to .
Assume that have the same left Wiener–Hopf factorization indices with respect to . By definition, and have the same left Wiener–Hopf factorization indices with respect to if and have the same left Wiener–Hopf factorization indices at infinity. This means that there exist matrices and such that . We have that We aim to prove that is invertible in and . Since the global invariant rational functions of and factorize in , and . Moreover, is a unit in as desired. Now, because and and factorize in . Therefore and are left Wiener–Hopf equivalent with respect to .
Conversely, let and such that Hence, . Put and . Therefore,
the global invariant rational functions of and of factorize in , and
the global invariant rational functions of and of factorize in
Then and are right equivalent (see the remark previous to Definition ▭). So, there exists such that Thus, . Since is biproper and is unimodular , have the same left Wiener–Hopf factorization indices at infinity. Consequentially, and have the same left Wiener–Hopf factorization indices with respect to .
In conclusion, for non-singular rational matrices with no zeros and no poles in the left Wiener–Hopf factorization indices with respect to form a complete system of invariants for the left Wiener–Hopf equivalence with respect to with .
A straightforward consequence of the above theorem is the following Corollary
Next we deal with the problem of factorizing or reducing a rational matrix to diagonal form by Wiener–Hopf equivalence. It will be shown that if there exists in an ideal generated by a monic irreducible polynomial of degree equal to 1 which is not in , then any non-singular rational matrix, with no zeros and no poles in admits a factorization with respect to . Afterwards, some examples will be given in which these conditions on and are removed and factorization fails to exist.
where are integers uniquely determined by . Moreover, they are the left Wiener–Hopf factorization indices of with respect to .
We prove now the uniqueness of the factorization. Assume that also factorizes as
with integers. Then,
The diagonal matrices have no zeros and no poles in (because ) and they are left Wiener–Hopf equivalent with respect to . By Theorem ▭, they have the same left Wiener–Hopf factorization indices with respect to . Thus, for all .
Following [5] we could call left Wiener–Hopf factorization indices with respect to the exponents appearing in the diagonal matrix of Theorem ▭. They are, actually, the left Wiener–Hopf factorization indices with respect to .
Several examples follow that exhibit some remarkable features about the results that have been proved so far. The first two examples show that if no assumption is made on the intersection and/or union of and then existence and/or uniqueness of diagonal factorization may fail to exist.
If is a polynomial matrix with zeros in then the existence of invertible matrices and such that with may fail. In fact, suppose that , . Therefore, and . Consider . Assume that with a unit in and a unit in . Thus, a nonzero constant and which is not a unit in .
If then the factorization indices with respect to may be not unique. Suppose that , with and , with a unit in and a unit in . Then can also be factorized as with a unit in and a unit in .
The following example shows that if all ideals generated by polynomials of degree equal to one are in then a factorization as in Theorem ▭ may not exist.
Notice that has no zeros and no poles in . We will see that it is not possible to find invertible matrices and such that
We can write with a unit in and . Therefore,
is invertible in and is also left Wiener–Hopf equivalent with respect to to the diagonal matrix .
Assume that there exist invertible matrices and such that , with integers. Notice first that is a nonzero constant and since and is a rational function with numerator and denominator relatively prime with , it follows that . Thus, . Let
From we get
As and , we can write and with , and . Therefore, by (▭), . Hence, or . In the same way and using (▭), or with a polynomial. Moreover, by (▭), must be non-negative. Hence, . Using now (▭) and (▭) and bearing in mind again that and , we conclude that and are polynomials.
We can distinguish two cases: , and . If and , by (▭), is a polynomial and since is proper, it is constant: . Thus . By (▭), . Since is polynomial and is proper, is also constant and then and . Consequentially, , and . This is impossible because is invertible.
If then , using (▭),
Notice that and unless . Hence, if , which is greater than . This cannot happen because is proper. Thus, . In the same way and reasoning with (▭) we get that is also zero. This is again impossible because is invertible. Therefore no left Wiener–Hopf factorization of with respect to exits.
We end this section with an example where the left Wiener–Hopf factorization indices of the matrix polynomial in the previous example are computed. Then an ideal generated by a polynomial of degree 1 is added to and the Wiener–Hopf factorization indices of the same matrix are obtained in two different cases.
which has a zero at 0. It can be written as with
where the global invariant factors of are powers of and the global invariant factors of are relatively prime with . Moreover, the left Wiener–Hopf factorization indices of at infinity are 3, 1 (add the first column multiplied by to the second column; the result is a column proper matrix with column degrees 1 and 3). Therefore, the left Wiener–Hopf factorization indices of with respect to are 3, 1.
Consider now and . There is a unimodular matrix , invertible in , such that is column proper with column degrees 3 and 2. We can write
where is the following biproper matrix
Moreover, the denominators of its entries are powers of and . Therefore, is invertible in . Since , the left Wiener–Hopf factorization indices of with respect to are 3, 2.
If , for example, a similar procedure shows that has as left Wiener–Hopf factorization indices with respect to ; the same indices as with respect to . The reason is that is not a divisor of and so with and as in (▭) and factorizing in .
5. Rosenbrock's Theorem via local rings
As said in the Introduction, Rosenbrock's Theorem ([1]) on pole assignment by state feedback provides, in its polynomial formulation, a complete characterization of the relationship between the invariant factors and the left Wiener–Hopf factorization indices at infinity of any non-singular matrix polynomial. The precise statement of this result is the following theorem:
Symbol appearing in (▭) is the majorization symbol (see [17]) and it is defined as follows: If and are two finite sequences of real numbers and and are the given sequences arranged in non-increasing order then if
with equality for .
The above Theorem ▭ can be extended to cover rational matrix functions. Any rational matrix can be written as where is the monic least common denominator of all the elements of and is polynomial. It turns out that the invariant rational functions of are the invariant factors of divided by after canceling common factors. We also have the following characterization of the left Wiener– Hopf factorization indices at infinity of : these are those of plus the degree of (see [4]). Bearing all this in mind one can easily prove (see [4])
Recall that for any rational matrix can be factorized into two matrices (see Proposition ▭) such that the global invariant rational functions and the left Wiener–Hopf factorization indices at infinity of the left factor of give the invariant rational functions and the left Wiener–Hopf factorization indices of with respect to . Using Theorem ▭ on the left factor of we get:
Theorem ▭ relates the left Wiener–Hopf factorization indices with respect to and the finite structure inside . Our last result will relate the left Wiener–Hopf factorization indices with respect to and the structure outside , including that at infinity. The next Theorem is an extension of Rosenbrock's Theorem to the point at infinity, which was proved in [4]:
Notice that Theorem ▭ can be obtained from Theorem ▭ when . In the same way, taking into account that the equivalence at infinity is a particular case of the equivalence in when , we can give a more general result than that of Theorem ▭. Specifically, necessary and sufficient conditions can be provided for the existence of a non-singular rational matrix with prescribed left Wiener–Hopf factorization indices with respect to and invariant rational functions in .
The proof of this theorem will be given along the following two subsections. We will use several auxiliary results that will be stated and proved when needed.
5.1. Necessity
We can give the following result for rational matrices using a similar result given in Lemma 4.2 in [18] for matrix polynomials.
It must be pointed out that may be an unordered -tuple.
with , , , , units in and factorizing in because has no poles and no zeros in . Therefore , where has as left Wiener–Hopf factorization indices at infinity and has as global invariant rational functions. Let . Hence,
with units in and factorizing in . Put . Its left Wiener–Hopf factorization indices at infinity are Lemma 2.3[4]. The matrix has as left Wiener–Hopf factorization indices at infinity. Now if then its invariant factors are , and, by Lemma 4.2[18],
Therefore, (▭) follows.
5.1.1. Proof of Theorem : Necessity
If are the invariant rational functions of in then there exist matrices invertible in such that
We analyze first the finite structure of with respect to . If , we can write as follows:
with and invertible matrices in . Thus are the invariant rational functions of with respect to . Let be the left Wiener–Hopf factorization indices of at infinity. By Lemma ▭ we have
As far as the structure of at infinity is concerned, let
Then and
where and are biproper matrices. Therefore are the invariant rational functions of at infinity. By Theorem ▭
Let (the symmetric group of order ) be a permutation such that and define , . Using (▭) and (▭) we obtain
for . When the previous inequalities are all equalities and condition (▭) is satisfied.
5.2. Sufficiency
Let be arbitrary elements such that . Consider the changes of indeterminate
and notice that . For , let denote the multiplicative subset of whose elements are coprime with . For as above define
In words, if () then
In general with equality if and only if . This shows that the restriction of to is a bijection. In addition is the restriction of to ; i.e.,
or .
In what follows we will think of as given elements of and the subindices of , and will be removed. The following are properties of (and ) that can be easily proved.
.
If then .
If is an irreducible polynomial then is an irreducible polynomial.
If are coprime polynomials then , are coprime polynomials.
As a consequence the map
with , is a bijection whose inverse is
where . In particular, if and (i.e. the complementary subset of in ) then
In what follows and for notational simplicity we will assume .
If factorizes in then factorizes in .
If is a unit of then is a unit of .
2. If is a unit of then it can be written as where and is a unit of . Therefore factorizes in . Since , it factorizes in (recall that we are assuming ) . So, is a unit of .
If and then .
If then .
If then .
If and then the matrix
1. Assume that and let be any element of . Therefore is a unit of and, by Lemma ▭.2, is a unit of . Moreover, is also a unit of . Hence, . Furthermore, if , it is a unit of and is a unit of .
2. If is any element of then and so . Since may divide we have that . Hence, and . Moreover if then , and . Thus, is a biproper rational function, i.e., a unit of .
3. If and is any element of then . Since and we obtain that . In addition, if , which is a unit of , then and since we conclude that is a unit of .
4. It is a consequence of 1., 2. and Remark ▭.2.
Notice that . Let , which is a non-zero constant, and put . Hence,
with
By 4 of Lemma ▭ matrices , and the Proposition follows.
5.2.1. Proof of Theorem : Sufficiency
Let be integers, irreducible rational functions such that , are monic polynomials factorizing in and integers such that and satisfying (▭).
Since and are coprime polynomials that factorize in and , by Lemmas ▭ and ▭, are irreducible rational functions with numerators and denominators polynomials factorizing in (actually, in ) and such that each numerator divides the next one and each denominator divides the previous one.
By (▭) and Theorem ▭ there is a matrix with as left Wiener–Hopf factorization indices with respect to and as invariant rational functions with respect to where , . Notice that has no zeros and poles in because the numerator and denominator of each rational function factorizes in and so it is a unit of .
Put and . As remarked in the proof of Proposition ▭, and . Now so that we can apply Proposition ▭ to with the change of indeterminate . Thus the invariant rational functions of in are .
On the other hand and so . Then we can apply Proposition ▭ to with so that is a non-singular matrix with no poles and no zeros in and as left Wiener–Hopf factorization indices with respect to . The theorem follows by letting .