Reduction to a Canonical Form of a Third-Order Polynomial Matrix with One Characteristic Root by means of Semiscalarly Equivalent Transformations

For the selected class of polynomial matrices of order three with one characteristic root with respect to the transformation of semiscalar equivalence, special triangular forms are established. The theorems of their uniqueness are proved. This gives reason to consider such canonical forms.


Introduction
In [1], it is proved that the matrix F(x) ∈ M(n, C[x]) of full rank by means of transformation: where P ∈ GL(n, C) and Q(x) ∈ GL(n, C[x]) can be reduced to the lower triangular form with invariant factors on the principal diagonal. Subdiagonal elements in a matrix of this form are ambiguously defined. e matrices F(x), G(x) which are related by the transformation (1) are called semiscalarly equivalent [1]. In [2], the specified triangular form for polynomial 3 × 3 matrices with one characteristic root is a little simplified. e resulting matrix of a simplified triangular form is called a reduced. In [2], the invariants of the reduced matrix are established. In particular, the invariance of the location of zero subdiagonal elements is proved. In [3], the reduced matrix, if there are some zero elements under its principal diagonal, by means of transformations of the form (1) (i.e., by means of semiscalarly equivalent transformations) is reduced to such matrices, which are uniquely defined. is gives grounds to consider the obtained matrices canonical for the selected class of matrices. is article introduces canonical forms for reduced matrices with all nonzero subdiagonal elements.

Previous Information
Here are some definitions and notations that will be used in this article, which are known from [2,3]. is applies to the definitions of the younger degree, of the younger term, of the younger coefficient, q-monomial, and q-coefficient of the polynomial and others. For example, monomial 4x 2 and its degree 2 are, respectively, a younger term and younger degree of polynomial f(x) � −3x 7 + 6x 5 − x 4 + 4x 2 , and 4 is the younger coefficient of this polynomial. Monomial 6x 5 and its coefficient 6 are, respectively, a 5-monomial and 5-coefficient of polynomial f(x). Let all the roots of the characteristic polynomial detF(x) (� characteristic roots) of the matrix F(x) be equal to each other; that is, the matrix F(x) has only one (without taking into account the multiplicity) characteristic root. Without loss of generality, we assume that the only characteristic root is zero and the first invariant factor of matrix F(x) is equal to one. With such assumptions, it is proved in [2] that, by means of semiscalarly equivalent transformations, the matrix F(x) is reduced to the matrix of the form which satisfies the following conditions: (4) Younger coefficients in a 1 (x) and a 2 (x) are units. e matrix A(x) of the form (2) with conditions (1)- (4) in [2] is called the reduced matrix. Next, we consider the situation where the last two invariant multipliers of the matrix A(x) do not coincide, that is, k 1 < k 2 . e case k 1 � k 2 was considered in [4]. e notation A(x) ≈ B(x) means that the matrices A(x) and B(x) are semiscalarly equivalent. It should be noted that the problem of classification with respect to semiscalar equivalence of matrices of the second order is solved in the article [5]. us, this article discusses other situations that differ from [4,5]. In [2], it is proved that, in case A(x) ≈ B(x), we can choose the left transformation matrix in the transition from A(x) to the reduced matrix of the lower triangular form. We will then apply semiscalarly equivalent transforms we can find the matrix B(x) and the right transformation Using the method of uncertain coefficients for given elements a 1 (x), a 2 (x), a 3 (x) and s 12 , s 13 , s 23 of matrices A(x) and S, respectively, with congruence we find b 1 (x) ∈ C[x], degb 1 < k 1 . We denote such elements by r(x) uv , u, v � 1, 2: r 11 (x) � 1 + s 12 a 1 (x) + s 13 a 3 (x), r 12 (x) � s 12 x k 1 + s 13 a 2 (x), Here . We form the matrix ‖r(x) uv ‖ 2 1 and consider the congruence with the unknown b 2 (x), b 3 (x). Since the free member of a matrix polynomial ‖r(x) uv ‖ 2 1 is a unit matrix, we can use the method of uncertain coefficients to solve this congruence and find . In addition to the above, we also denote By the above r ij (x) i, j � 1, 2, 3, and from the congruences (5) and (7) b i (x), we construct ‖r ij (x)‖ 3 1 and a matrix B(x) of the form (3), respectively. We can be convinced of equality SA(x) � B(x)‖r ij (x)‖ 3 1 . is means that ‖r ij (x)‖ 3 1 is reversible and its inverted matrix together with the matrix S reduces A(x) to B(x). If the matrix S (4) in the transition from A(x) to B(x) has one of the following views: then we will say that transformations of type I, transformations of type II, or transformations of type III, respectively, are applied to the matrix A(x). We shall use the following notation for matrices A(x) of form (2) and B(x) of form (3):

The Main Results
satisfy one of the following conditions: , which satisfies condition n j < k j , n j -monomial is absent, and in the first of these polynomials, which satisfies condition m j < k j , m j -monomial is absent.
(1) If 2q 3 ≥ k 2 , then A(x) is the desired matrix. Otherwise, we denote by d 0 and d 1 , respectively, the lower coefficient and the (2q 3 )-coefficient of the polynomial a 3 (x) and apply to A(x) transformations of the type III. In the left transformation matrix (see (9)), we put s 13 � d 1 /d 2 0 . e elements b i (x), i � 1, 2, 3, of the obtained in this way matrix B(x) satisfy the congruence: First, we obtain from (11) and (12) that the lower terms in b 1 (x), b 2 (x) are identical with the lower terms in b 1 (x), b 2 (x), respectively. Further note that the younger terms in δ B (x) and b 3 (x) coincide, the lower degrees of the last two additions in the lefthand side (13) exceed codegb 3 � q 3 , and inequality codeg( holds. erefore, by comparing the (2q 3 )-coefficients in both parts of (13), we obtain zero for (2) If in matrix A(x) has 2q 3 ≥ k 2 , then everything is proven-this matrix is the desired one. Otherwise, we will apply to it the transformation mentioned in Section 1. To show the absence of the (2q 3 )-mo- . e remaining considerations are the same as in paragraph 1. In order to not introduce new notations, we further assume that there is no , we apply transformation of the type II. At the same time, in the left transformation matrix (see (9)), we put s 12 � d 2 /d 0 , where d 0 and d 2 are, respectively, the lower coefficient and in this way satisfy the congruence: It can be seen from (14) and (15) that the younger terms in b 1 (x), b 2 (x) are the same as the lower terms in a 1 (x), a 2 (x), respectively (their coefficients are equal to one). Let us write (16) as follows: and codeg(a 1 (x)b 3 (x)) � q 1 + q 3 < 2q 1 + q 2 , then by comparing the (q 1 + q 3 )-coefficients in both parts of (17), we find that b 3 (x) contains no (q 1 + q 3 )-monomial. And because 2q 3

Journal of Mathematics 3
(3) Suppose that conditions q 3 > q 1 and q 3 < q 2 are satisfied in matrix A(x).
(1) If q 3 ≥ k 1 and 2q 3 ≥ k 2 , then all is proved-matrix A(x) is the desired one. (2) Let q 3 ≥ k 1 and 2q 3 < k 2 . Since q 3 < codega 2 , then q 3 (as well as q 1 ) is invariant (see Proposition 6 [2]). We apply to A(x) the transformation specified in Section 1. As a result, we obtain the matrix B(x) in the form (3). Its elements satisfy the congruences (11)-(13). Since q 1 + q 3 > k 1 , then from (11), we have a 1 (x) � b 1 (x). It can be seen from (12) that a 2 (x) � b 2 (x). Now we can represent (13) as From the last congruence, we have codegb 3 � q 3 . erefore, B(x) is a reduced matrix. If we take into account q 1 + q 2 > q 3 , then by comparing the (2q 3 )-coefficients at both times (19), we will conclude that, in b 3 (x), there is no (2q 3 )-monomial. If q 2 + q 3 ≥ k 2 , then everything is proved-matrix B(x) is the desired one. Otherwise, we take another step. In order not to introduce new notations, we will assume that element a 3 (x) of matrix A(x) does not contain (2q 3 )-monomial. We apply to A(x) transformations of the type I. In the left transformation matrix (see (9)), we put s 23 � d 2 /d 0 , where d 0 and d 2 are, respectively, the lower coefficient and the (q 1 + q 3 ) coefficient of polynomial a 3 (x). e elements b i (x), i � 1, 2, 3 of the resulting matrix B(x) satisfy the congruence: From (20), we have that a 1 (x) � b 1 (x), and from (21), it follows codega 2 � codegb 2 . en, from (22), we get where we can get codegb 3 � q 3 . Comparing the coefficients in both parts of (23), we conclude that there is no the monomial of degree q 1 + q 3 in polynomial b 3 (x). At the same time, in b 3 (x), as in a 3 (x), there is no (2q 3 )-monomial. (3) Now suppose that, in matrix A(x), we have q 3 < k 1 and 2q 3 ≥ k 2 . Apply to A(x) transformations of the type I. In the left transformation matrix (see (9)), we put s 23 � −d 3 /d 0 , where d 0 and d 3 are, respectively, the lower coefficient of the polynomial a 3 (x) and the q 3 -coefficient of the polynomial a 1 (x). As a result, we obtain a matrix B(x) of the form (3) whose elements b i (x), i � 1, 2, 3 satisfy the congruences (20)-(22). From (20), we have that codegb 1 � q 1 and q 3 -monomial in b 1 (x) is absent. From (21), codegb 2 � codega 2 , and from (22), we have codegb 3 � q 3 . It also follows from (20)-(22) that the lower coefficients in a i (x) and b i (x), i � 1, 2, 3 coincide. at is, B(x) is a reduced matrix. If q 1 + q 3 ≥ k 1 , then everything is already proven. en, B(x) is the desired matrix. Otherwise, in order to not introduce new notations, we consider the q 3 -coefficient in the element a 1 (x) of the matrix A(x) null. Denote by d 0 and d 4 , respectively, the lower polynomial coefficient of the polynomial a 3 (x)and (q 1 + q 3 )-coefficient of the polynomial a 1 (x). We perform over the matrix A(x) transformation of the type III. For this, we put s 13 � d 4 /d 0 in the left transformation matrix (see (9)). e elements of the resulting matrix B(x) satisfy the congruences (11)-(13). From (21), we obtain that codegb 1 � q 1 , the lower coefficient of the polynomial b 1 (x) is 1, and its q 3 -and (q 1 + q 3 )-coefficients are zero. From (12), it is seen that the lower coefficient in b 2 (x), as in a 2 (x), is equal to 1. erefore, the matrix B(x) has the necessary properties. (4) Let q 3 < k 1 and 2q 3 < k 2 . We can assume that the If this is not the case, then to A(x), we will apply transformation of the type I described in Section 3. If q 1 + q 3 < k 1 , then to A(x), we apply transformation of the type III described in Section 3. en, the resulting matrix will be zero (q 1 + q 3 )-coefficient and will remain zero q 3 -coefficient of the polynomial in position (2, 1). If q 1 + q 3 ≥ k 1 , then from the matrix A(x) by means of transformations of the type III referred to in item 1, we go to the redundant matrix B(x), in which 2q 3 -monomial of polynomial b 3 (x) is absent. en, q 3 -factor in b 1 (x) will also remain zero. is proves the first part of the theorem (existence).

Journal of Mathematics
Suppose that, in the reduced matrices A(x), B(x) of the forms (2) and (3), we have a 1 (x), a 2 (x), a 3 (x), b 1 (x), b 2 (x), b 3 (x) ≠ 0. Let us keep the notation given in theorem: . (31) We define polynomials: From the coefficients of each of the polynomials a 1 (x), a 3 (x), and a 11 (x), we form, respectively, columns a 1 , a 03 , and a 11 of height k 1 − q 1 . In the first place, in these columns, we put q 1 -coefficients, and below in order of increasing degrees, we place the rest of their coefficients, up to degree k 1 − 1 inclusive. We denote by a 2 , a 22 , and a 04 , the columns of height k 2 − k 1 − q 2 , constructed from the coefficients of polynomials a 2 (x), a 22 (x), and a 04 (x), respectively. In the first place in each of these columns, we put q 2 -coefficients. Below we place the rest of their coefficients (including zero) up to the degree k 2 − k 1 − 1. Similarly, from the coefficients of polynomials a 3 (x), a 32 (x), a 34 (x), and a 14 (x), we form columns a 3 , a 32 , a 34 , and a 14 and height k 2 − q 3 . Here, we also put in the first place q 3 -coefficients, and then, in the order of increasing degrees, we place all other coefficients. In the last places, there will be (k 2 − 1)-coefficients. For A(x), by the columns formed, we construct the matrices of the following form: K 1A � −a 03 0 a 11 � � � � � � � �, In complete analogy for B(x), we construct matrices of the following form: Obviously, in these matrices, each row consists of monomial coefficients of the same degrees. (2), we have a 1 (x), a 2 (x), a 3 (x) ≠ 0, q 3 > q 1 , q 3 > q 2 and n 1 :

Theorem 2. Let in the reduced matrix A(x) of the form
In addition, one of the following conditions is true: (1) In b 1 (x), n 1 -monomial is absent, if n 1 < k 1 and n 2 < k 2 . (2) In b 2 (x), (codegδ A + k 1 )and n 2 -monomials are absent, if n 1 ≥ k 1 and n 2 < k 2 . (3) In b 1 (x), q 3 -and n 1 -monomials are absent, if n 1 < k 1 and n 2 ≥ k 2 . (4) In the first column of the matrix K B (35), the coefficients of the polynomials b 1 (x), b 2 (x), b 3 (x) are zero elements that correspond to the maximum system of the first linearly independent rows of the submatrix K 0B , if n 1 ≥ k 1 and n 2 ≥ k 2 .
e matrix B(x) is uniquely defined.
Proof. Existence. Let n 1 < k 1 . We apply to A(x) transformation of the type II with the left transformation matrix of the form (9). At the same time, we put s 12 � d 1 /d 0 , where d 0 is the younger coefficient and d 1 is the n 1 -coefficient in a 3 (x). e elements b i (x), i � 1, 2, 3, of the thus obtained reduced matrix B(x) satisfy the congruences (14)-(16). We write (16) in the form where r 21 (x) � a 1 (x) − b 1 (x) − s 12 a 1 (x)b 1 (x)/x k 1 ∈ C[x]. Comparing the n 1 -coefficients in both parts of the last congruence, we have that b 3 (x) does not contain n 1 -monomial. We further assume that element a 3 (x) of the matrix A(x) does not contain n 1 -monomial (if n 1 < k 1 ).

Conclusion
e matrices B(x), whose existence is established in eorems 1 and 2, can be considered canonical in the class of semiscalarly equivalent matrices. e method of their construction follows from the proof of the first parts of these theorems. is completes the study of semiscalar equivalence of third-order polynomial matrices with one characteristic root, started in the previous works of the author. e results obtained in this article, as well as the results of the works cited here, are applicable to the study of the simultaneous similarity of sets of numerical matrices. In this context, the works of [6][7][8][9] should be noted. ese results also have utility in solving Sylvester-type matrix equations over polynomial rings. Such equations often arise in applied problems.

Data Availability
Data from previous studies were used to support this study.
ey are cited at relevant places within the text as references.

Conflicts of Interest
e author declares that there are no conflicts of interest.