Abstract
The equivalence of systems plays a critical role in multidimensional systems, which are usually represented by the multivariate polynomial matrices. The Smith form of a matrix is one of the important research contents in polynomial matrices. This paper mainly investigates the Smith forms of some multivariate polynomial matrices. We have obtained several new results and criteria on the reduction of a given multivariate polynomial matrix to its Smith form. These criteria are easily checked by computing the minors of lower order of the given matrix.
1. Introduction
The subject of multidimensional systems is concerned with a mathematical framework for tackling a broad range of paradigms whose analysis or synthesis requires the use of functions and polynomials in several complex variables. Many physical systems, multiple-input multiple-output systems, data analysis procedures, and learning algorithms have a natural structure due to the presence of one spatial variable. So the theory of systems is widely applied in areas of image processing, linear multipass processes, geophysical exploration, iterative learning control systems, etc. [1–9]. The equivalence of systems is one important research problem in the system theory. It is often required to transform a given system into a simpler but equivalent form. As we all know, a multivariate polynomial matrix is often used to represent an system. So the equivalence problem of system is often transformed into the equivalence problem of polynomial matrices. For 1D systems, the equivalence problem has been solved [5, 7] by the quite mature theory of 1D polynomial matrix. For () case, since the equivalence problem is equivalent to a highly difficult problem, the isomorphism problem of two finitely presented modules, there is no hope that the equivalence problem can be solved completely. There exist two primary problems on equivalence of multivariate polynomial matrices: one is to reduce an polynomial matrix to its Smith form. Kung et al. have obtained some interesting results about the equivalence of polynomial matrices to their Smith forms [5, 6, 10–13]. Furthermore, the Smith forms of some polynomial matrices can be computed by Maple [14]. The other is called Serre’s reduction problem. One of the motivations for doing Serre’s reduction for an polynomial matrix is to reduce an system to an equivalent system containing fewer equations and unknowns. Cluzeau et al. have studied Serre’s reduction and presented some new interesting results in [15, 16].
The following problem, proposed by Serre in 1960s, plays an important role in the research problems of systems. It is not only the problem of reducing a matrix to its Smith form, but also Serre’s reduction problem.
Problem 1. When is an polynomial matrix equivalent to the matrixwhere ? is the identity matrix and is the zero matrix.
For the real number field , Lin et al. [12] have investigated Problem 1 for with and proved that is equivalent to its Smith form. Li et al. [17] have investigated Problem 1 for with and obtained that is equivalent to its Smith form with satisfying some criteria. Cluzeau et al. [13–16] also studied it and gave some new interesting results.
In this paper, we will investigate Problem 1 for the case of and det , where are nonnegative integers, is an arbitrary field (even a function field or a finite field). Then, we investigate the Smith forms of some rectangular polynomial matrices and consider the following problem.
Problem 2. Let with , where is the greatest common divisor of the minors of , are nonnegative integers. When is equivalent to its Smith form?
The paper is organized as follows. In Section 2, we give some basic concepts on the equivalence of polynomial matrices. In Section 3, main results and some tractable criteria on equivalence of several kinds of polynomial matrices are proposed. In Section 4, an example is provided to illustrate the effectiveness of our constructive method.
2. Preliminaries
In the following, will denote the polynomial ring in variables with coefficients in arbitrary field , will be an algebraic closed field of , will be the dimensional vector space over , will denote the set of matrices with their entries in , will denote the set of matrices with their entries in , the identity matrix will be denoted by and the zero matrix will be denoted by . A matrix over with its determinant in is said to be unimodular. Throughout the paper, the argument is omitted whenever its omission does not cause confusion.
Definition 1. (see [18]).Let be of full row (column) rank. Then is said to be zero left prime (zero right prime) if the minors of generate the unit ideal .
If is zero left prime (zero right prime), then is called simply to be ZLP (ZRP). According to the Quillen-Suslin theorem [19], we have that is ZLP if and only if there is a matrix such that . It is also equivalent to say that any ZLP (ZRP) matrix over can be completed to an unimodular (invertible) matrix.
Definition 2. Let , , and be polynomially defined as follows:where is the rank of , , is the greatest common divisor of the minors of , and satisfies , . Then, we define the Smith form of as
Definition 3. (see [17]). Let and denote two matrices in , , and are said to be equivalent if there exist two invertible matrices and such that .
3. Main Results
In this section, the main results are presented. First, we give some well-known results and provide an answer to Problem 1 for case of in Subsection 3.1. Then we extend this result to more general case of and present a complete answer to Problem 2 in Subsection 3.2.
3.1. Equivalent Theorem
In order to prove our main results, we first give several useful lemmas.
Lemma 1. (see [20]). Let , then have no common zeros in (are zero coprime) if and only if there exist such thati.e., is a ZLP row vector in , or generate the unit ideal .
Lemma 2. (see [20]). Let and . Suppose that , then is a divisor of .
Lemma 3. (see [17]). Let , . If the minors of have no common zeros in (generate ), then the minors of have no common zeros in (generate ) for .
Proof. The proof is similar to that of Lemma 2.2 in [17], so it is omitted here.
Lemma 4. (see [18]). Let be of normal full rank, if the reduced minors of generate , then there exists a ZLP matrix such that .
The following result is presented in [20], for the convenience of the reader, we record it here.
Lemma 5. (see [20]). Let , , if the minors of generate , then the minors of also generate .
Proof. Assume that the minors of are . If the minors of generate , then there exist such thatSubstitute , thenIt is straightforward that are the minors of . Thus the minors of also generate .
Lemma 6. Let , then and the minors of have no common zeros if and only if the minors of have no common zeros.
Proof. Necessity. Assuming that the minors of have a common zero point , by Laplace expansion, we can easily know that is also a zero point of , contradicting the fact that and the minors of have no common zeros. Thus, the minors of have no common zeros.
Sufficiency. It is straightforward.
For the convenience, we first define as follows:We have the following key conclusion which is very important to derive our main results.
Lemma 7. Let and , where is unimodular. If all the minors of generate , then is equivalent to its Smith form .
Proof. We prove this by induction on . When , letthenObviously, . Note that is unimodular, then have no common zeros. Assume that have a zero , where , then has a common zero . This is a contradiction. Thus, have no common zeros, and is a unimodular row. According to Quillen-Suslin Theorem, there exists a unimodular matrix such thatSowhere . Then there exists the unimodular matrix such thatNote that , . Thus, is equivalent to its Smith formSo the conclusion is true for .
Assume that the conclusion is true for , we investigate the case of . Let , where , , , . ThenLet be the first row of . Since is unimodular, using Laplace Theorem and expanding its first row, we obtain that , where are all the minors of and . Combined with Lemma 1, we have that have no common zeros. Assume that have a common zero , then all the minors of have a common zero . This is a contradiction. Thus, is a unimodular row in , by the Quillen-Suslin Theorem, there exists a unimodular matrix such thatSowhere , . Then there exists the matrixsuch thatSetting , since all the minors of have no common zeros, combined with Lemma 3, we have that all the minors of have no common zeros. Note that an minor of is just an minors of or or 0, then and the minors of have no common zeros. By Lemma 6, the minors of have no common zeros. By the inductive hypothesis, there exist two unimodular matrices such thatthenThus, is equivalent to the matrix , combined with is equivalent to the matrix ; then we obtain that is equivalent to the matrix , and is the Smith form of .
Now we are going to state one of our main results, which give partial answer to Problem 1. We recall the notation
Theorem 1. Let with . is equivalent to its Smith form if and only if all the minors of generate .
Proof. Sufficiency. Because , then the rank of . Since all the minors of generate , by Lemma 5, the minors of also generate , then for every . By Lemma 4, there exists a ZRP column vector such thatBy the Quillen-Suslin theorem, an unimodular matrix can be constructed such that is its last column. Then the elements of the last column of are zero polynomials. By Lemma 2, the last column of have the common divisor , i.e.,for some . Let , then we haveNote that are unimodular, and , thenFrom Lemma 3, we have that the minors of also generate . Note that , combined with Lemma 5, we obtain that the minors of also generate for . Similarly, a unimodular matrix can be constructed such thatwhere . Let , then we haveNote that is unimodular, and , then , is unimodular, andBy Lemma 7, the matrix is equivalent toCombined with are unimodular, we obtain that is equivalent to its Smith form .
Necessity. If is equivalent to , there exist two unimodular matrices such that . Note that all the minors of have no common zeros; combined with Lemma 3, we have that all the minors of have no common zeros, i.e., all the minors of generate .
In the following, we will extend the above result to more general case.
3.2. Generalization
Lemma 8. Let with , , and be a unimodular matrix. If all the minors of generate and are positive integers, then is equivalent to .
Proof. Letwhere , , , .
Thenand .
Next, we will proof that is a ZLP matrix. Note that is unimodular, then is a ZLP matrix. Let denote all the minors of . Then all the minors of are . We can prove that have no common zeros. Suppose that have a common zero and combined with have no common zeros, so is a zero of and . Note that the elements of the last row of matrix all have the factor and that an minor of is just or includes the factor , so the minors of have the common zero , this is a contradiction. Thus, all the minors of have no common zeros, and it is a ZLP matrix. According to Quillen-Suslin theorem, there exists an unimodular matrix such thatThen,for some . Let Then,Note that are unimodular, then , thus is equivalent to .
Now we investigate Problem 1 for the case of . Denote
Theorem 2. Let with , where is a positive integer. Then is equivalent to its Smith form if and only if all the minors of generate .
Proof. Sufficiency. Notice that , we have that rank . Since all the minors of generate the unit ideal , according to Lemma 5, then the minors of also generate the unit ideal . Hence, for every , repeating the proof of Theorem 1, we have thatwhere , are unimodular, with . By Lemma 3, all the minors of also generate . According to Lemma 7, there exist unimodular matrices such that . ThusAgain by Lemma 3, we obtain that all the minors of generate . If , iterating the preceding process, we obtain thatwhere with . ThenLet , we have that is unimodular and all the minors of generate . By Lemma 8, is equivalent to , that is, there exist unimodular matrices such that . Furthermore,Let , , then are unimodular, andIf , iterating the same procedure successively, we obtain thatwhere are unimodular. Note that , then ; that is, is a unimodular matrix. So is equivalent to its Smith form .
Necessity. It is straightforward that all the minors of generate . According to Lemma 3, all the minors of also generate .
Next we investigate Problem 1 for the case of and det , where are nonnegative integers.
Theorem 3. Let with , where are nonnegative integers. If all the minors of generate , then is equivalent to its Smith form
Proof. (1) If , then , i.e., is unimodular, so is equivalent to .
(2) If or , we have that or . Without loss of generality, we assume that . Because , then the rank of . Since all the minors of generate , by Lemma 5, the minors of also generate ; hence, for every . By Lemma 4, we obtain a ZRP column vector such thatAccording to the Quillen-Suslin Theorem, an unimodular matrix can be constructed such that is its last column. Then the last column of are zero polynomials. By Lemma 2, the last column of have the common divisor , i.e.,for some . Let , then we obtain thatBy Lemma 3, all the minors of also generate the unit ideal . If , iterating the preceding process, we obtain thatFurthermore,By Lemma 3, the minors of generate the ideal , according to Lemma 8, there exist unimodular matrices such that . ThenFurthermore, , combined with Lemma 3, the minors of also generate .
If , iterating the same procedure successively, we obtain thatwith . Thus, is equivalent to its Smith form .
(3) If are positive integers, then or . Without loss of generality, we assume that . By Theorem 2, combined with the conclusion of the case (2) above, we have that is equivalent to the matrix . In the following, we prove that is equivalent to the matrix
Letwhere , , , and .
Thenwhere , .
In fact, we can prove that is a ZLP matrix. Note that is unimodular, then is a ZLP matrix. Let denote all the minors of . Then all the minors of are . We can prove that have no common zeros. Suppose that have a common zero , note that have no common zeros, so and have the common zero . Moreover, the elements of the last row of matrix have the common factor , since an minors of is just or includes the factor , so the minors of have the common zero , this is a contradiction. Thus, all the minors of have no common zeros, and it is a ZLP matrix. By the Quillen-Suslin theorem, there exists an unimodular matrix such thatThen,for some . There exists a unimodular matrix such thatNote that are unimodular, then ; thus, is equivalent to . Note that is equivalent to the matrix , combined with the definition of the Smith form of a matrix, we obtain that is equivalent to its Smith form .
Theorem 3 provides a positive answer to Problem 1 for the case of and det . It also gives a sufficient condition to check this kind of matrices are equivalent to their Smith forms; in fact, this condition is also a necessary condition.
Theorem 4. Let with det , where are non-negative integers, then is equivalent to its Smith formif and only if all the minors of generate .
Proof. Sufficiency. From Theorem 3, it is straightforward.
Necessity. By computing, we can easily obtain that all the minors of generate . By Lemma 3, all the minors of generate .
Remark 1. In the theorem above, is an arbitrary field. When is the real field and , Theorem 4 is same as Theorem 2.5 in Li et al. [17]. Furthermore, if and , Theorem 4 is just Proposition 4 in Lin et al. [12]. So Theorem 4 extends the above two results.
In the following, we will investigate the Smith forms of some rectangular polynomial matrices and consider Problem 2. Let denote the greatest common divisor (g.c.d) of the minors of the matrix , denote the g.c.d of all the minors of and denote
Theorem 5. Let () be of full row rank and suppose , where are nonnegative integers. Then is equivalent to its Smith formif and only if all the minors of generate .
Proof. Sufficiency. According to Theorem 3.3 in [21], there exists , such that , where , and is a ZLP matrix. Combined with Theorem 5, we can obtain two unimodular matrices such that . ThenIt is obviously that is also ZLP. According to the Quillen-Suslin Theorem, we can construct an unimodular matrix such that is its first rows. Henceand . Since , combined with Lemma 3, we have that , then the Smith form of is .
Necessity. If is equivalent to its Smith form , we see that all the minors of generate . Using Lemma 3, we obtain that all the minors of generate .
Remark 2. Let be nonzero polynomials, a necessary and sufficient condition for generates that is the reduced Gröbner basis of the ideal generated by includes a unit in the field . According to Theorems 4 and 5, we can check whether an polynomial matrix is equivalent to its Smith form by using the existing Gröbner basis algorithms to the ideal generated by the lower minors of . Thus, the conditions of Theorems 4 and 5 can be verified easily.
4. Example and Algorithm
In this section, we first present a 2D example to illustrate our result and explain how to obtain the unimodular matrices associated with the equivalence of system matrices in the method. Then we give an algorithm to deal with the equivalence of the kind of matrices we discussed to their Smith forms.
In many areas of engineering such as Circuits and Signals, the general 2D systems can be defined in terms of the generalized Rosenbrock system [8] aswhere is the state vector, is the input vector, is the output vector, , , , are polynomial matrices, is a field. The operators and may have various meanings depending on the type of system. For example, in delay-differential systems, represents a differential operator and a delay-operator. For 2D discrete systems, and represent horizontal and vertical shift operators, respectively. This system gives rise to the system matrix in the general form:
Example 1. Consider a system matrixwhereBy computing, , the minors of is as follows: , , , , , , , , .
The reduced Gröbner basis of the ideal generated by is . So the minors of generate unit ideal , by Theorem 4, is equivalent to its Smith form. ConsiderSolving the equations, , where , we obtain , where is the transpose of . It then follows thatWe complete into a unimodular matrix such that , whereWe then havewhereWe consider againRepeating the procedure above for , we have thatwhereThen,By Theorem 3.5 in [17], we have that is equivalent to , andwhereSo,Consider , factor it like above, we obtain thatwhereThen,We consider againSolving the equations, , where , we obtain , where is the transpose of . It then follows thatWe complete into a unimodular matrix such that , whereWe obtain thatwhereThus,Now we consider . By Lemma 7, we have that is equivalent to . AndwhereThus,Repeating the procedure above for , we obtain that . Furthermore, we consider and obtain . SowhereSettingwe obtain thatLet , , we haveNow we consider the matrix . From Theorem 2, we know that is equivalent to . AndwhereThen we haveLet , now consider the matrix . By Theorem 3, we have that is equivalent to , wherewhereNowLet , thenwe haveBy computing, = = 1.
Thus is equivalent to the matrixNow we can find , , such thatwhere , , , .
Thus, the systemis equivalent to the systemwhich is simpler.
With the help of Example 1 and Theorem 3, we now can get the Algorithm 1.
The program of the function SyzygyModule which we use in algorithm can be found in https://faculty.math.illinois.edu/Macaulay2/doc/Macaulay2-1.15/share/doc/Macaulay2/MCMApproximations/html/_syzygy__Module.html.
Another function CompleteMatrix is available in http://wwwb.math.rwth-aachen.de/QuillenSuslin. Moreover, for a new algorithm for CompleteMatrix, see the package MatrixFactorization in http://www.mmrc.iss.ac.cn/dwang/software.html, which contained a ZLP algorithm. This algorithm can obtain a unimodular matrix whose inverse is the complete matrix of a given ZLP matrix.
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5. Conclusions
In this paper, we have investigated the equivalence problem of several kinds of polynomials matrices over an arbitrary field, and have presented some interesting results. We have obtained some criteria for these matrices to equivalent to their Smith forms respectively. These criteria are easily checked by the existing Gröbner basis algorithm for the ideal generated by the minors of lower order of a given matrix. We also give an example to illustrate our method. All of these could provide useful information for engineers to reduce systems.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors are grateful to Professor Sang-Bing Tsai and the anonymous referees for their numerous instructive comments, which have significantly improved the presentation of the paper. This research is supported by the National Natural Science Foundation of China (11871207) and the Graduate Research and Innovation Project of Hunan Province in China (CX2018B674).