Abstract
This paper solves the problem of classifying matrices over a ring of polynomials up to semiscalarly equivalence transformations. For the selected set of polynomial matrices of simple structure, the so-called oriented by characteristic roots reduced matrix is constructed. The latter, in addition to the triangular form and the presence of invariant factors on the main diagonal, has some predefined properties. Invariants and conditions of semiscalar equivalence are established for such matrices.
1. Introduction
Let be the field of complex numbers and the ring of polynomials in an indeterminate over . Let , denote the algebras of -by- matrices over , , respectively, and , their corresponding groups of units (general linear groups). Two matrices , of are said to be semiscalarly equivalent when there exists a pair such that (see [1, 2])
This is an equivalence relation; its equivalence classes are called semiscalarly equivalence classes. Therefore, there is a problem of classification of polynomial matrices up to semiscalar equivalence. If a matrix has a full rank, then in the class there is a lower triangular matrix in which its invariant factors stand on the main diagonal (see [1] Theorem 1 §1 Section 4). A similar result was also obtained in [3, 4]. The specified triangular matrix is defined by the class ambiguously; therefore, it cannot be used at once for the establishment of belonging of any matrix to this class. Thus, there is a need to specify in class such a matrix, which is determined with a lesser degree of ambiguity and for which we can find a system of invariants. To do this, it is necessary to clarify the mentioned triangular form of the matrix, that is, to establish such a matrix, which, in addition to the triangular form and the presence on the main diagonal of its invariant factors, has some predetermined properties. The problem of classifying of polynomial matrices of the same size up to semiscalar equivalence contains the matrix pair problem, which is the problem of classifying a pair of square matrices of the same size up to similarity transformations. Donovan and Freislich [5] call a matrix problem “wild” if it contains the matrix pair problem and “tame” otherwise. A certain characterization of time and wild problems is given by Drozd [6]. Thus, the problem of classifying polynomial matrices up to semiscalar equivalence is wild.
We investigate the problem of determining when two matrices are semiscalarly equivalent. This study aims to construct a simpler form in the class of semiscalarly equivalent matrices and to find its invariants. In this paper, we limit ourselves to the case when a matrix has a simple structure, a complete rank, and a unit first invariant factor. The latter condition is not significant. More precisely, all the obtained results can be applied to a wider class of matrices. It should also be noted that this article is a continuation of the research initiated by the author [7]. Much useful information on the issues discussed in this article can be found in the monograph [8]. In particular, this monograph also investigates other types of equivalences of matrices and finite sets of matrices over different rings. Some generalizations of the concept of semiscalar equivalence in the case of matrices over quadratic rings and their application to the solution of matrix equations can be found in recent work [9].
2. Preliminary Information
Suppose that in the class (see (1)), we have a matrixwhere , , divides and , that is, , and are invariant factors. Since has a simple structure, polynomials , do not have multiple roots. We assume that . Otherwise, we have or . Then, the task is slightly simplified and is the subject of another study. Denote by and the sets of roots of polynomials and , respectively. The notation means divides , and is the greatest common divisor of polynomials and . The corresponding notation will be used in the case of three polynomials. Let also be for element of matrix . In the future, we will denote the zero element of the main field and zero rows and columns as . We can assume that in the matrix for some fixed root, is satisfied the condition
This is easily achieved by adding some multiples with the numerical coefficients of the first row of a matrix to its other rows.
Proposition 1. The greatest common divisor for the class (see (1)) does not depend on the choice of the matrix with condition (3) with a fixed characteristic root .
Proof. Suppose, in addition to , to the class belongs some other matrixwhere , , .
Then, equalityis satisfied for some , . If we put in the latter equality, then we have , so . Also from (5), you can writeSince , then . Therefore, from (6) and (7), we have that . If we take into account the symmetry of semiscalar equivalence, then we have that . Therefore, .
If , then . This case is the subject of labor research [7]. In this article, we consider the situation when . Then, there is a root such that .
3. Reduction to a Special Triangular Form
Proposition 2. If for matrix (2) of class condition (3) and are satisfied, then in this class, there is such a matrix (4) that and , , for some .
Proof. If in for a fixed we have , then the desired matrix is achieved by adding some multiple (with a numerical coefficient) of the second row of the matrix to its third row. If , , then first we choose so that for every . Then, from the congruencesconsistently find , , of degrees , , . Next, from elements , , and we construct a matrix of the form (4) and matriceswhereIt is easy to see that and . By direct verification, we establish that the matrices , and together with the original matrix satisfy equality (5). This means that belongs to the class and also , . By adding to the third row of the matrix its second row multiplied by , we obtain the matrix we need with zero value of the (3, 1)-element at .
We can further assume that for the matrix referred to in Proposition 2.
Proposition 3. If the elements of matrices (2), (4) of class satisfy the conditions:for some , then and the matrix in relation (5) has an upper triangular form.
Proof. The proof of Proposition 1 shows that in the matrix from equality (5), there is . If we put in (7), we have . Then, from (7), it is seen that since . Given the symmetry of the relation of semiscalar equivalence, we can obtain that . Therefore, .
Proposition 4. If for elements of matrices (2), (4) of class are true conditions (8) for some roots , and , then the following conditions are also satisfied:(i), ;(ii)Equivalence is performed for arbitrary ;(iii)If , then for such that , .
Proof. If we take into account that , then from (6), we can writewhere do we get the first part of the condition (i) Putting in (12) , we have, respectively,Since , then from (13), (14), we obtain that for an arbitrary . Also from (14), (15) follows such equivalence . This fully proves conditions (i) and (ii).
Equality based on the proved (i), (ii) is obvious for those that . In the case of , this equality follows from (13)–(15), if we take into account .
Corollary 1. Suppose that the conditions of Proposition 4 for matrices (2), (4) are satisfied, and letThen, for such that , , .
Proof. It is easy to see that , . Under Proposition 4, we have . Therefore, .
Proposition 5. Suppose that for matrix (2), the conditions of Proposition 4 hold. Let also and be a fixed and an arbitrary root of the set , respectively, for which both values are different from 0 and 1, and . Then, in the class there is a matrix (4) with the conditions of Proposition 4 and an arbitrarily fixed value other than and (16).
Proof. Let us choose for an unknown polynomial at some value other than and . Consider the equationwith the unknown for each element specified in Proposition, where is defined in (9). Let us write the latter in this formSince (see the proof of Corollary 1), it is solvable for each element stated in Proposition. Let us now construct a polynomial of degree by its values for each root as follows:Consider the equationrelative to . If we take into account the equality for every specified in Proposition, and the inequality , then the rank of the matrix of this equation is 2, and it has a solution with nonzero all its components. Based on this solution, we write the congruenceand consider the equationwith unknown , whereFrom (22), we can find polynomials , of degrees , since . From the found , and we construct a matrix B (x) of the form (4) and a matrixWe also construct a matrix , in which the upper left -block coincides with the above matrix and, in addition, , ,It is easy to see that the matrix together with the constructed , non-singular and invertible satisfy equality (5). In addition, the element of the matrix has a preselected value . Therefore, is the desired matrix.
Proposition 6. Suppose that for matrices (2), (4) with the conditions of Proposition 4, we have for some andThen, in these matrices , , , and in the upper triangular matrix from relation (5), we have , .
Proof. For the elements of the matrices , the congruence (12) is performed. Putting in it and , we obtain, respectively, equality (13) andFrom them, it immediately follows that , , because the values are different from and . Then, if we return to (12), we obtain . If we take into account what has already been proved, then from (5), we can write the congruences as follows:from which it followsFrom (28), (29), and (30), it is easy to see that , and , respectively.
Corollary 2. For matrices (2), (4) of class with the conditions of Proposition 6, we have the following equivalences:
Proposition 7. Suppose that for the elements of matrices (2), (4), the conditions of Proposition 4 are satisfied, and let , are defined in Proposition 6. Then equivalencesare fulfilled for such that .
Proof. Matrices , satisfy equation (5), where the matrix according to Proposition 3 has an upper triangular form. From (5), except (12), we obtain the congruences as follows:If we exclude from (33) and (34) the members containing s13, s23, s33, we getFrom (12) for we have (13), and from (35) for and , respectively, we obtainFrom (13) and (36) follows the first of the required equivalences, and from (36) and (37)-the second.
Proposition 8. Suppose that in the matrix (2) with the conditions of Proposition 4, element of the set acquires only the values and . Let also be for some , where , are defined in Proposition 6. Then, in the class there is a matrix of the form (4) with the conditions of Proposition 4, in which and .
Proof. Denote , , and construct matricesObviously, and . We also construct a matrix B (x) of the form (4), in which and elements of degrees are defined from the congruenceBased on the elements of matrix and elements ,we construct matrix . Obviously, . We make sure that the constructed matrices , and together with satisfy equality (5). Therefore, , like , belongs to class . The elements of the matrices , satisfy the congruence (35), where , and , are defined above. If we put in this congruence, we get .
Proposition 9. Suppose that for matrices (2), (4) of the class, conditions of Proposition 8 are satisfied, and we have , for some . Then, the conclusion of Proposition 6 is true.
Proof. Congruences (13) and (35) are performed for the elements of matrices , . If in these congruences to put accordingly and , then from the received, we will receive that and . The rest of the proof is the same as in the proof of Proposition 6.
Corollary 3. For matrices (2), (4) with the conditions of Proposition 9, we have and .
4. Main Results
Definition 1. Matrix of Proposition 4, in which or for each and or for each , is called oriented by the characteristic roots reduced matrix. Matrix of Proposition 5, for whichis called oriented by the characteristic roots reduced matrix. Matrix of Proposition 8 is called oriented by the characteristic roots , reduced matrix.
Theorem 1. Suppose that in the oriented by the same characteristic roots or , reduced matrices (2), (4), we have , andwhere (notation see Proposition 6). Matrices , are semiscalarly equivalent if and only if their elements coincide, and the rest satisfy the following conditions:(i), , ;(ii)for arbitrary such roots that , we have(iii)for arbitrary roots we have(iv)for arbitrary roots such that and rows , are pairwise different, there is a number such that
Proof. (i)Necessity(ii)For semiscalarly equivalent matrices , the equality , condition (i) and condition (ii) for zero values , follow from Propositions 6 or 9. For the elements of matrices , the congruences (33) and (34) are satisfied, from which (35) follows. If we consider that and (see Propositions 6 or 9), then from (35), we have From the latter equivalence for () and nonzero values , it follows the first equality from (43). The second of them follows from the first, since , , .(iii)Equivalence (43) for zero values and follows from (i). If and , then from (33) for we have whence . Since in this case , , so . This proves the equivalence (44) for zero values and nonzero values . Consider now the case where and . If we take into account (43), then from (34), where , we have whence . Then, it is easy to trace the chain of implications(iv)Suppose we have three roots , for which (respectively, ). For elements of matrices , the congruences (28), (29) hold. From (28) and (29) follows the congruence (30). From (29) or (30), depending on whether or , for , , we obtainwhere . Since , thenIf we take into account that and (see (43)), then from (57), we have equalityfrom which it follows equality (45). From conditionfollows either , , or , , . Thenand , , or , . Since , then in each case, from system (51), we havewhere it follows (43).
Sufficiency. Let are all those elements from the set for which and all rows are pairwise different. Under the conditions of theorem (see (iii)), the rows are also pairwise different, and each has a nonzero first element. Consider the equationwith the unknown . If we take into account that and , , (see condition (ii)), then from (45), it follows that in the matrix of equation (57), every three rows are linearly dependent. Hence, this equation has a nonzero solution. If for some , , we havethen from conditions (i), (ii) can be obtainedThen, it follows from (46) that in the matrix of equation (57), the rows for the same are linearly dependent. Therefore, this equation has a solution with a nonzero first component . Based on this solution, taking into account conditions (i)–(iii), we can write the congruenceswhere , , , . It should be borne in mind that for each . It is clear that from (60) and (61), it is possible to pass to (29), (30). If we take into account that and , then by adding to the congruence (30) multiplied by the congruence (29), we proceed to the congruence (28) with the same above-mentioned coefficients . Based on the elements of matrices , and elements , we construct matriceswhereIt is easy to verify the reversibility of matrix and the truth of equality (5) for constructed , and the original matrices , . The theorem is proved.
Theorem 2. Suppose that in the oriented by the same characteristic roots the reduced matrices (2), (4) (see Definition 1) we have\where is defined in Proposition 6. Matrices , are semiscalarly equivalent if and only if their elements coincide, and the rest satisfy the following conditions:(i), , ;(ii)for arbitrary roots or we have ;(iii)for arbitrary roots or such that are nonzero and pairwise different, we have
Proof. Necessity. According to Proposition 4, the elements of matrices , on the set acquire the same values (0 and 1). Since , then .(i)Matrices , satisfy equality (5), in which the matrix according to Proposition 3 has an upper triangular form. For the elements of these matrices from (5) follow the congruences (33), (34). If we exclude from them, we can obtain a congruence (30). From the latter, given the symmetry of the relation of semiscalar equivalence, it follows . Therefore, for every . This means that for the same . Also from (34), for we have whence . Therefore, . If we put in (33) and take into account that , , we can write If you put and take into account then from (33), we have From the obtained equalities (67), (68), it follows that , , whence .(ii)The required equivalence for zero values follows from (i), because then the values are also zero. Let now . From (34) for we have and for we have and (66). From each of the obtained pairs of equalities, it follows that .(iii)Let or and all values are nonzero and pairwise different. Under conditions (i), (ii), are also nonzero and pairwise different. If we put in (66), then in each of the cases or we get systemsorrespectively. Since , then from each of the obtained systems, we havewhence it follows (65).
Sufficiency. Let in matrices the elements coincide, and for the rest conditions (i)–(iii) are fulfilled. If , then (see definition) and . Then, from (i), it follows that . In this case, and everything is proved. Now let and is the maximum subset of the set for which all values are nonzero and if or , . Obviously, . Under conditions (i), (ii), we have and . Let also , , . Consider the equationrelative to the unknown . The rank of the matrix of this equation does not exceed 4, because, as follows from (iii), the rank of each of the selected submatrices is not more than 2. This means that the equation has a nonzero solution. Moreover, it is easy to see that there is a solution , where . Based on this solution, the congruences can be written:where , , , , . Since , , for every (see definition and condition (i)), (75) can be written aswhere . From the congruences (76) and (77), we can writesince all are different andFor or under conditions (i), (ii), the following implications are true:Therefore, the congruence (78) is performed modulo , that is, the congruence (34) is true. Since , for every and for every , we can move from (75) to congruenceand from it to the congruence (30), if we take into account (80). If we add to the congruence (30), the congruence (34) multiplied by and take into account that , , then we can obtain the congruence (33). According to the above , , , and elements of matrices , we construct matriceswhereThe first of these matrices is not singular. In the second matrix , since , and the equal polynomials on the set acquire only two values (0 and 1). We also have , considering (33) and (34). We can see that and moreover, both of these matrices , together with , satisfy equality (5). This means that , are semiscalarly equivalent. This completes the proof of the theorem.
5. Conclusion
In the class of semiscalarly equivalent polynomial matrices of simple structure, the so-called oriented by characteristic roots reduced matrix is established. It has a lower triangular form, invariant factors on the main diagonal, and some other properties. The left transformation matrix in the transition from one oriented by characteristic roots reduced matrix to another semiscalarly equivalent oriented (by the same characteristic roots) reduced matrix has an upper triangular form (Propositions 3, 6, 9). Theorems 1 and 2 give the necessary and sufficient conditions of the semiscalarly equivalence of oriented by the same characteristic roots reduced matrices. Proof of all statements is constructive. Based on them, we can specify the method of constructing the left and right transformation matrices in the transition from an arbitrary triangular matrix with invariant factors on the main diagonal to the oriented by characteristic roots reduced matrix (Propositions 2, 5, 8). This makes it possible to apply the obtained results to the solution of linear two-sided matrix equations of the type with unknown nonsingular matrix over and invertible over . Matrix equations of this type are often found in mathematical physics, other sections of mathematics, and applied problems.
Data Availability
The data used to support the findings of the study are included within the article as references.
Conflicts of Interest
The author declares that there are no conflicts of interest.