Oriented by Characteristic Roots ReducedMatrices in the Class of Semiscalarly Equivalent

A set of polynomial 3 × 3-matrices of simple structure has been singled out, for which a so-called oriented by certain characteristic roots reduced matrix is established in the class of semiscalarly equivalent. *e invariants of such reduced matrices and the conditions of their semiscalar equivalence are indicated. *e obtained results will also be applied to the problem of similarity of sets of numerical matrices.


Introduction
By eorem 1 (see Section 4 in [1] and [2]), matrix F(x) ∈ M(n, C[x]) of full rank by transformation where P ∈ GL(n, C) and Q(x) ∈ GL(n, C [x]) are reduced to the lower triangular form with invariant factors on the main diagonal. We consider the case when F(x) is a 3 × 3-matrix of simple structure and its first invariant factor is equal to one. e latter condition does not limit the generality in the sense that everything stated in this article can be extended to 3 × 3-matrices, which after reduction (i.e., division of all their elements) by the first invariant factor become matrices of simple structure. By the definition of a matrix of simple structure, all its elementary divisors are linear. is concept is taken from [1], where it is introduced for polynomial matrices of arbitrary degree by analogy with the concept of numerical matrices (above the field) of simple structure with [3]. Note that the numerical matrices of a simple structure in [4] are called simple and in [5] are nondefective. Next, we will use the terminology from [1,2], namely, the transformation (1) and the matrices F(x) and G(x) will be called semiscalarly equivalent (abbreviation: ssk.e., notation: F(x) ≈ G(x)). Our task is to construct in the class PF(x)Q(x) { }a lower triangular matrix with invariant factors on the main diagonal and some predefined properties, to find a complete system of its invariants and to establish on this basis the conditions of ssk.e. for two such matrices. is problem for 2 × 2-matrices in the general case is solved by the author in [6] and for 3 × 3-matrices with one characteristic root in [7].

In class PF(x)Q(x)
{ }, there is a matrix of the form where dega 1 (x) < degϕ 1 (x), dega 2 (x), dega 3 (x) < degϕ 2 (x), and ϕ 1 (x) divides a 2 (x) and ϕ 2 (x). Let us denote ϕ 12 (x): � ϕ 2 (x)/ϕ 1 (x) and a 2 ′ (x): � a 2 (x)/ϕ 1 (x). We assume that degϕ 1 { } can be chosen so that a 1 (x) ≡ 0 or a 2 (x) ≡ 0. In these cases, the task of finding the conditions ssk.e. matrices are greatly simplified. Hereinafter, the notation M 1 and M 2 will mean the sets of roots of polynomials ϕ 1 (x) and ϕ 12 (x), respectively. Recall that element α ∈ M 1 ∪ M 2 is called the characteristic root of the matrix A(x) and M 1 ∪ M 2 for class PF(x)Q(x) { } is defined unambiguously (i.e., does not depend on the choice of matrix A(x)). Obviously, A(x) in class PF(x)Q(x) { } can be chosen so that, for some α 0 ∈ M 1 , the following condition is satisfied. In the future, this condition will be mandatory for A(x) and for all triangular matrices of class Hereinafter, record (a(x), b(x)) will mean the greatest common divisor of polynomials a(x) and b(x). We will use the corresponding notation for the greatest common divisor of three polynomials.
We divide the set M 2 of roots of polynomial ϕ 12 (x) into subsets according to the following principle: we assign el- we get some partition of the set M 2 : Proof. Suppose that, in addition to A(x), the matrix B(x) (4) with a condition similar to (3) also belongs to class (5) holds, from which we have (9) and Let α 1 and α 2 belong to one of the subsets of (10), for example, to K 1 . If x � α 1 and x � α 2 is put alternately and the obtained equations are subtracted in (9), then we get e last two equations are written in matrix form If, in (10), m � 1, then the elements a 2 (x) and a 3 (x) in A(x) are determined to the nearest multiplier, and then, A(x) can be chosen so that a 2 (x) ≡ 0. In this case, A(x) can be considered canonical in class PF(x)Q(x) { }. erefore, we consider the case when m > 1.
}. e proposition is proved.
Denote by N 1 and N 2 the subsets of the set M 2 , for elements β k ∈ N 1 and c l ∈ N 2 of which we have a 3 (β k ) ≠ 0 and a 2 (c l ) ≠ 0, respectively. Propositions 2 and 5 imply the following.
{ } for fixed β 1 , β 2 ∈ M 2 do not depend on the choice of the matrix A(x), which satisfies the conditions of Proposition 5.
Since we have imposed the condition that, in (10), m > 1, then N 1 is not an empty set. If N 2 is empty, then a 2 (x) ≡ 0. In this case, the problem of ssk.e. for 3 × 3-matrices is reduced to a similar problem for 2 × 2-matrices. e latter, as mentioned in Section 1, is resolved in [6]. erefore, in the future, we assume that N 2 is not an empty set.
Let A(x) satisfy the conditions of Proposition 5. It is clear that one of the values a 3 (β 1 ) and a 3 (β 2 ) is nonzero. Let a 3 (β 1 ) ≠ 0. We can also assume that, in A(x), one of the following conditions is fulfilled: Each of these conditions for A(x) is achieved by multiplying the first two rows and the first two columns by the corresponding numerical factors. Such a matrix A(x) is called oriented by characteristic roots β 1 , β 2 , β 3 reduced matrix. Proposition 6. If the intersection N 1 ∩ N 2 for oriented by the same characteristic roots β 1 , β 2 , β 3 reduced matrices A(x) (2) and B(x) (4)

of class PF(x)Q(x)
{ } is not empty, then s 11 � s 22 in the matrix ‖s ij ‖ 3 1 of equation (5).
Proof. According to Proposition 5, the matrix ‖s ij ‖ 3 1 in question is an upper triangular, and s 12 � 0. Equations (9) and (12) hold for elements of matrices A(x) and B(x). From them, we can get a congruence: If we put, in (13), x � β 3 , we immediately get s 11 − s 22 � 0. e proposition is proved. (2) and B(x) (4) be oriented by the same characteristic roots β 1 , β 2 , β 3 reduced matrices and

Conditions of Semiscalar Equivalence of Reduced Matrices
If, for the sets N 1 and N 2 defined in the corollary, the intersection N 1 ∩ N 2 is not empty, then the matrices A(x) and B(x) are ssk.e. if and only if the following conditions are met: or for each β k ∈ N 1 and c l ∈ N 2 . If the intersection N 1 ∩ N 2 is empty, then the matrices A(x) and B(x) are ssk.e. if and only if the following conditions are met: (2) If a 3 (β i ) ≠ 1 for some β i ∈ N 1 , then for every β k ∈ N 1 , and if a 2 ′ (c j ) ≠ 1 for some c j ∈ N 2 , then for each c l ∈ N 2 .
Proof (necessity). Let A(x) ≈ B(x). en, for the elements of matrices A(x) and B(x) are equations (9) and (12), in which according to Proposition 6 in the absence of intersection of sets N 1 and N 2 we have s 11 � s 22 . erefore, from (13), we immediately have (i). If in equations (9) and (12) put x � β 1 , x � β 3 , x � β k , and x � c l , where β k and c l mean an arbitrary element of N 1 and N 2 , respectively, the result can be written in the matrix form as follows: Hence, we have From the last equality (ii), since s 11 , For arbitrary β k ∈ N 1 and c l ∈ N 2 , we have each equality with (14), since s 11 , s 33 ≠ 0. If Ω A (c j ) ≠ 0 for some c j ∈ N 2 , then, for any β k ∈ N 1 and c l ∈ N 2 , similarly we have each of equation (15). Now, let the sets N 1 ,N 2 not intersect and β k ∈ N 1 and c l ∈ N 2 . If we put x � β 1 , x � β k and x � β 3 , x � c l , respectively, in (9) and (12), we obtain equations that can be written as Hence, we have Since s 11 , s 22 , s 33 ≠ 0, then, from the last equality, we obtain (j). If a 3 (β i ) ≠ 1, for some β i ∈ N 1 , then, from equality, We get (16) for each β k ∈ N 1 . Similarly, if a 2 ′ (c j ) ≠ 1, for some c j ∈ N 2 , then, from equality, We obtain (17) for every c l ∈ N 2 . e necessity is proved. characteristic roots reduced matrix (Proposition 4). e matrix of this special shape is lower triangular and has invariant factors on the main diagonal and some predefined properties. Propositions 1-3 and 5 and Corollary indicate the invariants of the reduced matrix. Propositions 5 and 6 set out the form of the left transform matrix, which together with some right transform (polynomial) matrix translates one reduced matrix into another within class PF(x)Q(x) { }. is matrix has an upper triangular form with a zero element in position (1,2). In some cases, it has the same first two diagonal elements. e theorem specifies the necessary and sufficient conditions under which the reduced matrices are ssk.e. us, Propositions 1-3 and 5, corollary, and theorem indicate a complete system of invariants of the reduced matrix with respect to ssk.e. It is clear that theorem can be applied to establish ssk.e for arbitrary matrices F(x) and G(x) (not reduced) if they belong to the set selected in this paper. Namely, F(x) and G(x) are ssk.e. if and only if their oriented by the same characteristic roots reduced matrices A(x) and B(x)satisfy the conditions of theorem. Also, as already mentioned in Section 1, everything described in this article can be extended to 3 × 3-matrices, which after division by the first invariant factor become matrices of simple structure. e proved theorem can be applied to the problem of classification of finite sets of numerical matrices with accuracy to similarity. In particular, let sets F and G of numerical (from the field) 3 × 3-matrices be given. On the matrices of these sets, as on the coefficients, we construct matrix polynomials F(x) and G(x) with single higher coefficients. Suppose that, after division by the first invariant factor of the polynomial matrices F(x) and G(x) obtained in this way, we have particles F′(x) and G ′ (x) of simple structure.
en, for the similarity of sets F and G, it is necessary and sufficient to satisfy the conditions of theorem for consolidated matrices A(x) and B(x) such that A(x) ≈ F ′ (x) and B(x) ≈ G ′ (x).

Data Availability
e data used to support the findings of the study are given within the article as references.

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