Reduced Triangular Form of Polynomial 3-by-3 Matrices with One Characteristic Root and Its Invariants

Abstract

In this paper the semiscalar equivalence of polynomial matrices is investigated. We introduce the notion of the so-called reduced triangular form with respect to semiscalar equivalence for the 3-by-3 matrices with one characteristic root and indicate the invariants of this reduced form.

1. Introduction

We consider the following equivalence relation in the set of all polynomial matrices of fixed order over the field of complex numbers: matrices are called semiscalarly equivalent if there exist invertible matrices over and , respectively, such that [1] (see also [2]); notation . Several other notions of the equivalence (so-called PS- equivalence) of the polynomial matrices are considered in [3]. Two matrices and are said to be PS- equivalent if there exist , with . If are semiscalarly equivalent (or PS- equivalent), then they must have the same characteristic roots and the same invariant factors. By Theorem 1 [1] (see also Theorem 1 §1, Section IV [2]) every matrix of full rank is semiscalarly equivalent to the lower triangular form with invariant factors on the main diagonal. The similar results can be found in [4]. However, the matrix of this form is not uniquely defined. Therefore, the question when two matrices are semiscalarly equivalent is open. The conditions of semiscalar equivalence of order 2 polynomial matrices in [5–7] are indicated. In this paper is determined so-called reduced form with respect to semiscalar equivalence for the 3-by-3 matrices with one characteristic root and its invariants are found. The problem of semiscalar equivalence (as of PS- equivalence) contains the classical linear algebra problem of reducing a pair of numerical matrices to a canonical form by a simultaneous similarity transformation (for the solution of this problem, see [8]).

Let . We assume that characteristic polynomial has a unique root. Without loss of generality, we assume that uniquely characteristic root is zero and the first invariant factor of the matrix is unit. In accordance with [1] at this assumption we havewhere , (divides). We consider , since the case is considered in [9].

2. Preliminary Results

Proposition 1. In the class of semiscalarly equivalent matrices there exists a matrix of the form (1), in which , , , , .

Proof. Proof is obvious.

Let the matricesbe given, where , , , , , , , .

Proposition 2. A left reducible matrix in the passage from to the semiscalarly equivalent of the form (2) is an upper triangular matrix.

Proof. Let . Then, we have where , . From (3) it follows that Substituting in (4), (5), we find that . Since , the right-hand side of equality (6) and the second summand of the left-hand side of this equality are divisible by . Therefore, . This implies that . Proposition is proved.

Proposition 3. If for the matrices (2), then , , where the bracket denotes the greatest common divisor.

Proof. Since in equality (3), it follows that, from (4) and (5), where , we obtain and , respectively. Thus, . The notation of semiscalar equivalence is a symmetric relation, so that . The first part of the Proposition is thus proved. Similarly, from (5) and (6), where , we can obtain and , respectively. Therefore, . Again by virtue of symmetrical relation of semiscalar equivalence we obtain . The Proposition is proved completely.

Further, by using semiscalarly equivalent transformations , we reduce the matrix to a matrix of the form (2) with the predefined properties. Furthermore, the left reducible matrix , obviously, must be selected of the upper triangular form. We shall show how by the given matrix and by the left reducible matrix we can find the matrix of the form (2) and the right reducible matrix such that . Then, we shall choose the matrix of the upper unitriangular form:By the given entries and of the matrices and , respectively, by means of the method of indeterminate coefficients from the congruence we find , . Denote by , , such entries:Here, . Construct the matrix and consider the congruence in the unknowns . This congruence is solvable, since the free term of the matrix polynomial is a nonsingular matrix. The unknowns can be found by the method of the indefinite coefficients. It is easily verified that . Besides the above definition of , , let us introduce the following notations:By the indicated above entries , , and by the definition from congruence (8), (10) we construct the matrix and the matrix of the form (2). Make sure that the equality is valid. This means that the matrix is invertible and its inverse matrix with reduces to . If the matrix (7) in the passage from to has one of the following formsthen we shall say that to the matrix is applied the transformation of type I or the transformation of type II, respectively.

3. Improvement of the Triangular Form of Matrix in the Class of Semiscalarly Equivalent Matrix: Reduced Matrix

Junior degree of polynomial , , is the least degree of the monomial (of nonzero coefficient) of this polynomial; notation . The monomial of degree and its coefficients are called the junior term and junior coefficients, respectively. Denote by symbol the junior degree of the polynomial .

Proposition 4. If in the matrix of the form (2) , then , where in the matrix of the form (2) , , .

Proof. We will uniquely determine the value of from condition and we will apply to the matrix the transformation of the type I. As a result we obtain the matrix of the form (2). Its entries , , satisfy the congruences:where . From (14), (15), and (13), we find that , and , respectively. Proposition is proved.

Proposition 5. Let a matrix of the form (2) be given such that , . Then there exists a matrix of the form (2) such that and , , , where .

Proof. By transformation of type II we reduce the matrix to the matrix of the form (2). Herewith in the matrix (see (12)) we define such that the inequality is true. The entries of the obtained matrix satisfy the congruences:where . From (16) and (18) we have that and , respectively. If the principle of a choice of is considered, then from (17) it follows that . Proposition is proved.

Proposition 6. Let the matrix have the form (2) and where . Then there exists a matrix of the form (2) such that and , , .

Proof. Let us apply to the matrix the transformation of type II. Moreover, in the left reducible matrix (see (12)) we can choose the value of so that the condition is fulfilled. As a result we obtain the matrix of the form (2) in which its entries , , satisfy the following congruences:From (19) we have that . Then (20) implies and from (22) we findFrom (21) and (23) by excluding of , we arrive at the congruenceorSince , from (21) we have . From the inequalities and it follows that junior terms in both members (25) coincide with the junior term of the product . Then from (23), taking into account the choice of , we find . Proposition is proved.

Proposition 7. Let a matrix of the form (2) be given in which and . Then , where has the form (2) in which , and in the monomial of degree is absent.

Proof. Let us apply to the matrix a transformation of type II. In this case, in the left reducible matrix (see (12)) we define from the condition , where and are junior coefficient and coefficient of the monomial of degree in , respectively. Then the entries and of the matrix and of the matrix , obtained as a result of transformation, satisfy the congruences (16) – (18). From (16) it follows at once that junior terms in , coincide and in the monomial of degree is absent. From (17) and (18) we have, respectively, that and . Proposition is proved.

Taking into account Propositions 4 and 5 we shall think henceforth that and in the matrix of the form (2), if . If , then based on Propositions 6 and 7, we note that in the matrix the inequality holds true and in the monomial of degree is absent. Moreover, we may take the junior coefficients of the polynomials and to be unit, if . If one of the polynomials is identical zero, then we may take the junior coefficients of the nonzero underdiagonal entries of the matrix to be unit. Such matrix we shall call the reduced matrix. All subsequent semiscalarly equivalent transformations of the matrix should not violate her property to be reduced.

4. Invariants of the Reduced Matrix

Theorem 8. In reduced matrix of the form (2) , , and are invariants with respect to semiscalarly equivalent transformations.

Proof. Let and be reduced matrices of the form (2) and . From equality (3), where matrix by Proposition 2 is upper triangular, we get Recall that . If , then from (26) it follows that , i.e., . Let (). If , then from (26) at once we have . If , then . In view of Proposition 3, we get Also by Proposition 3 we have . For this reason and from (26) it follows that . Thus, .
From equality (37) we can writeWe recall that . If , then (29) implies that . Then, from (28) we find , since , .
Let (). If , then (29) implies that , and from (28), taking into account the form of and , we have .
If , then from (28) we get also . If , then from (26) we obtain , and . Therefore, from (29) it is clear that . In particular, . If , then from (26) we have . In this case, rewrite (29) in the detailed form as where , . Since and , as seen from the last congruence, . Moreover, , if . Also in this case from last congruence we obtain . Theorem is proved.

Corollary 9. Let in the reduced matrix one of the following three conditions hold true:Then left reducible matrix in the passage from to the reduced matrix is of the formif condition (31) is fulfilled, or if one from two conditions (32), (33) is valid.

Corollary 10. Identical equality to zero of the entry , , or of the reduced matrix is invariant with respect to semiscalarly equivalent transformations.

Remark. If some two underdiagonal entries in the reduced semiscalarly equivalent matrices , are nonzero, then diagonal entries of the left reducible matrix, which by Proposition 2 is upper triangular, are equal to each other. Therefore, we can choose this matrix as unitriangular.

Let reduced matrices , of the form (2) be given. Henceforth we shall apply the following notations:

Corollary 11. If and in the reduced matrix , then . Therefore, is an invariant with respect to semiscalarly equivalent transformations.

Theorem 12. In the reduced matrix the quantity is an invariant with respect to semiscalarly equivalent transformations.

Proof. Let , be reduced matrices of the form (2) and . Then from equality (3), where matrix is upper triangular (see Remark), we obtainwhere . Excluding from (37) and (38) the summand, which contains , we define Since , , from (39) it follows that .

Corollary 13. The congruence is an invariant of with respect to semiscalarly equivalent transformations.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.