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

Authors: B. Z. Shavarovskii
Journal: Journal of Mathematics
 (2018)

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## 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.


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## Body

## 1. Introduction

We consider the following equivalence relation in the set of all polynomial matrices of fixed order over the fieldC of complex numbers: matrices F(x),G(x) are called semiscalarly equivalent if there exist invertible matrices P,Q(x) over C and C[x], respectively, such that G(x)=PF(x)Q(x) [1] (see also [2]); notation F(x)≈G(x). Several other notions of the equivalence (so-calledPS- equivalence) of the polynomial matrices are considered in [3]. Two matrices F(x) and G(x) are said to bePS- equivalent if there exist P(x)∈GL(n,C[x]), Q∈GL(n,C) with G(x)=P(x)F(x)Q. If F(x),G(x) are semiscalarly equivalent (orPS- 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 ofPS- 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]).LetF(x)∈M(3,C[x]). We assume that characteristic polynomial detF(x) has a unique root. Without loss of generality, we assume that uniquely characteristic root is zero and the first invariant factor of the matrix F(x) is unit. In accordance with [1] at this assumption we have(1)Fx≈Gx=100g1xxk10g3xg2xxk2,where k1≤k2, xk1|g2(x) (divides). We consider k1<k2, since the case k1=k2 is considered in [9].


## 2. Preliminary Results

Proposition 1.
In the class{PF(x)Q(x)} of semiscalarly equivalent matrices there exists a matrix of the form (1), in which deg⁡g1<k1, deg⁡g2,deg⁡g3<k2, g1(0)=g3(0)=0, g2(x)=xk1g2′(x), g2′(0)=0.Proof.
Proof is obvious.Let the matrices(2)Ax=100a1xxk10a3xa2xxk2,Bx=100b1xxk10b3xb2xxk2be given, where k1<k2, deg⁡a1,deg⁡b1<k1, deg⁡a2,deg⁡b2,deg⁡a3,deg⁡b3<k2, a1(0)=a3(0)=b1(0)=b3(0)=0, a2(x)=xk1a2′(x), a2′(0)=0, b2(x)=xk1b2′(x), b2′(0)=0.Proposition 2.
A left reducible matrix in the passage fromA(x) to the semiscalarly equivalent B(x) of the form (2) is an upper triangular matrix.Proof.
LetA(x)≈B(x). Then, we have (3)sij13100a1xxk10a3xa2xxk2=100b1xxk10b3xb2xxk2rijx13,where sij13∈GL(3,C), rij(x)13∈GL(3,C[x]). From (3) it follows that (4)s21+s22a1x+s23a3x=b1xr11x+xk1r21x,(5)s31+s32a1x+s33a3x=b3xr11x+b2xr21x+xk2r31x,(6)s32xk1+s33a2x=b3xr12x+b2xr22x+xk2r32x.Substituting x=0 in (4), (5), we find that s21=s31=0. Since xk1|r12(x), the right-hand side of equality (6) and the second summand of the left-hand side of this equality are divisible by xk1+1. Therefore, xk1+1|s32xk1. This implies that s32=0. Proposition is proved.Proposition 3.
If A ( x ) ≈ B ( x ) for the matrices (2),then(a1(x),a3(x),xk1)=(b1(x),b3(x),xk1), (a2(x),a3(x),xk2)=(b2(x),b3(x),xk2),where the bracket(,,)denotes the greatest common divisor.Proof.
Sincer11(0)≠0 in equality (3), it follows that, from (4) and (5), where s21=s31=s32=0, we obtain (a1(x),a3(x),xk1)|b1(x) and (a1(x),a3(x),xk1)|b3(x), respectively. Thus, (a1(x),a3(x),xk1)|(b1(x),b3(x),xk1). The notation of semiscalar equivalence is a symmetric relation, so that (b1(x),b3(x),xk1)|(a1(x),a3(x),xk1). The first part of the Proposition is thus proved. Similarly, from (5) and (6), where s33≠0, we can obtain (b2(x),b3(x),xk2)|a2(x) and (b2(x),b3(x),xk2)|a3(x), respectively. Therefore, (a2(x),a3(x),xk2)|(b2(x),b3(x),xk2). Again by virtue of symmetrical relation of semiscalar equivalence we obtain (a2(x),a3(x),xk2)|(b2(x),b3(x),xk2). The Proposition is proved completely.Further, by using semiscalarly equivalent transformationsA(x)→SA(x)R(x)=B(x), we reduce the matrix A(x) to a matrix B(x) of the form (2) with the predefined properties. Furthermore, the left reducible matrix S, obviously, must be selected of the upper triangular form. We shall show how by the given matrix A(x) and by the left reducible matrix S we can find the matrix B(x) of the form (2) and the right reducible matrix R(x) such that A(x)≈B(x)=SA(x)R(x). Then, we shall choose the matrix S of the upper unitriangular form:(7)S=1s12s1301s23001.By the given entries a1(x),a2(x),a3(x) and s12,s13,s23 of the matrices A(x) and S, respectively, by means of the method of indeterminate coefficients from the congruence (8)a1x+s23a3x≡b1x1+s12a1x+s13a3xmodxk1we find b1(x)∈C[x], deg⁡b1<k1. Denote by ruv(x), u,v=1,2, such entries:(9)r11x=1+s12a1x+s13a3x,r12x=s12xk1+s13a2x,r21x=a1x+s23a3x-b1xr11xxk1∈Cx,r22x=1+s23a2′x-s12+s13a2′xb1x.Here, a2′(x)=a2(x)/xk1∈C[x]. Construct the matrix ruv(x)12 and consider the congruence (10)b3xb2xruvx12≡a3xa2xmodxk2in the unknowns b2(x),b3(x). This congruence is solvable, since the free term of the matrix polynomial ruv(x)12 is a nonsingular matrix. The unknowns can be found by the method of the indefinite coefficients. It is easily verified that b2′(x)=b2(x)/xk1∈C[x]. Besides the above definition of ruv(x), u,v=1,2, let us introduce the following notations:(11)r13x=s13xk2,r23x=s23xk2-k1-b1xr13x,r31x=a3x-b3xr11x-b2xr21xxk2∈Cx,r32x=a2x-b3xr12x-b2xr22xxk2∈Cx,r33x=1-b3xr13x-b2xr23xxk2∈Cx.By the indicated above entries rij(x), i,j=1,2,3, and by the definition from congruence (8), (10) bi(x) we construct the matrix rij(x)13 and the matrix B(x) of the form (2). Make sure that the equality SA(x)=B(x)rij(x)13 is valid. This means that the matrix rij(x)13 is invertible and its inverse matrix with S reduces A(x) to B(x). If the matrix S (7) in the passage from A(x) to B(x) has one of the following forms(12)10001s23001or1s120010001,then we shall say that to the matrix A(x) 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 degreeof polynomial f(x)∈C[x], f(x)≠0, is the least degree of the monomial (of nonzero coefficient) of this polynomial; notation codeg⁡f. The monomial of degree codeg⁡f and its coefficients are called thejunior term andjunior coefficients, respectively. Denote by symbol +∞ the junior degree of the polynomial f(x)≡0.Proposition 4.
If in the matrixA(x) of the form (2) codeg⁡a1=codeg⁡a3≠+∞, then A(x)≈B(x), where in the matrix B(x) of the form (2) codeg⁡b1>codeg⁡b3, codeg⁡b2=codeg⁡a2, codeg⁡b3=codeg⁡a3.Proof.
We will uniquely determine the value ofs23 from condition codeg⁡(a1(x)+s23a3(x))>codeg⁡a1 and we will apply to the matrix A(x) the transformation of the type I. As a result we obtain the matrix B(x) of the form (2). Its entries bi(x), i=1,2,3, satisfy the congruences:(13)a1x+s23a3x-b1x≡0modxk1,(14)a2x-b2x1+s23a2′x≡0modxk2,(15)a3x-b2xr21x-b3x≡0modxk2,where r21(x)=(a1x+s23a3x-b1x)/xk1∈C[x]. From (14), (15), and (13), we find that codeg⁡b2=codeg⁡a2, codeg⁡b3=codeg⁡a3 and codeg⁡b1>codeg⁡b3, respectively. Proposition is proved.Proposition 5.
Let a matrixA(x) of the form (2) be given such that codeg⁡a2′=codeg⁡a3≠+∞, a2′(x)=a2(x)/xk1∈C[x]. Then there exists a matrix B(x) of the form (2) such that A(x)≈B(x) and codeg⁡b1=codeg⁡a1, codeg⁡b3=codeg⁡a3, codeg⁡b2′>codeg⁡b3, where b2′(x)=b2(x)/xk1∈C[x].Proof.
By transformation of type II we reduce the matrixA(x) to the matrix B(x) of the form (2). Herewith in the matrix S (see (12)) we define s12 such that the inequality codeg⁡(s12xk1a3(x)-a2(x))>codeg⁡a2 is true. The entries b1(x),b2(x),b3(x) of the obtained matrix B(x) satisfy the congruences:(16)b1x-a1x1-s12b1x≡0modxk1,(17)b2x+s12xk1a3x-a2x1+s12a1x≡0modxk2,(18)b3x-a3x1-s12b1x-a2xr21x≡0modxk2,where r21(x)=(b1x-a1x1-s12b1x)/xk1∈C[x]. From (16) and (18) we have that codeg⁡b1=codeg⁡a1 and codeg⁡b3=codeg⁡a3, respectively. If the principle of a choice of s12 is considered, then from (17) it follows that codeg⁡b2′>codeg⁡b3. Proposition is proved.Proposition 6.
Let the matrixA(x) have the form (2) and (19)+∞≠2codeg⁡a1+codeg⁡a2′=codeg⁡a3≥codeg⁡a2,where a2′(x)=a2(x)/xk1∈C[x]. Then there exists a matrix B(x) of the form (2) such that A(x)≈B(x) and b1(x)=a1(x), codeg⁡b2=codeg⁡a2, codeg⁡b3>codeg⁡a3.Proof.
Let us apply to the matrixA(x) the transformation of type II. Moreover, in the left reducible matrix (see (12)) we can choose the value of s12 so that the condition codeg⁡(a3(x)+s12a12(x)a2′(x))>codeg⁡a3 is fulfilled. As a result we obtain the matrix B(x) of the form (2) in which its entries bi(x), i=1,2,3, satisfy the following congruences:(20)a1x-b1x-s12a1xb1x≡0modxk1,(21)a2x-b2x+s12b1xb2x-xk1b3x≡0modxk2,(22)a3x+b1xb2′x-b3x1+s12a1x-b2′xa1x≡0modxk2.From (19) we have that 2codeg⁡a1≥k1. Then (20) implies b1(x)=a1(x) and from (22) we find(23)a3x-b3x+s12a1xb1xb2′x-b3x≡0modxk2.From (21) and (23) by excluding of s12, we arrive at the congruence(24)a1xa2x-xk1a3x≡b1xb2x-xk1b3xmodxk2+codeg⁡a1,or(25)a1xa2′x-a3x≡b1xb2′x-b3xmodxk2-k1+codeg⁡a1.Since codeg⁡(a1(x)a2(x)-xk1a3(x))=codeg⁡(a1(x)a2(x)), from (21) we have codeg⁡b2=codeg⁡a2. From the inequalities codeg⁡(a1(x)a2′(x))<k2-codeg⁡a1 and k2-k1+codeg⁡a1≥k2-codeg⁡a1 it follows that junior terms in both members (25) coincide with the junior term of the product a1(x)a2′(x). Then from (23), taking into account the choice of s12, we find codeg⁡b3>codeg⁡a3. Proposition is proved.Proposition 7.
Let a matrixA(x) of the form (2) be given in which codeg⁡a3≥codeg⁡a2 and 2codeg⁡a1<k1. Then A(x)≈B(x), where B(x) has the form (2) in which codeg⁡b2=codeg⁡a2, codeg⁡b3≥codeg⁡b2 and in b1(x) the monomial of degree 2codeg⁡b1=2codeg⁡a1 is absent.Proof.
Let us apply to the matrixA(x) a transformation of type II. In this case, in the left reducible matrix (see (12)) we define s12 from the condition s12c02=c1, where c0 and c1 are junior coefficient and coefficient of the monomial of degree 2codeg⁡a1 in a1(x), respectively. Then the entries ai(x) and bi(x) of the matrix A(x) and of the matrix B(x), obtained as a result of transformation, satisfy the congruences (16) – (18). From (16) it follows at once that junior terms in a1(x), b1(x) coincide and in b1(x) the monomial of degree 2codeg⁡a1 is absent. From (17) and (18) we have, respectively, that codeg⁡b2=codeg⁡a2 and codeg⁡b3≥codeg⁡b2. Proposition is proved.Taking into account Propositions4 and 5 we shall think henceforth that codeg⁡a3≠codeg⁡a1 and codeg⁡a3≠codeg⁡a2′ in the matrix A(x) of the form (2), if codeg⁡a3<codeg⁡a2. If codeg⁡a3≥codeg⁡a2, then based on Propositions 6 and 7, we note that in the matrix A(x) the inequality codeg⁡a3≠2codeg⁡a1+codeg⁡a2′ holds true and in a1(x) the monomial of degree 2codeg⁡a1 is absent. Moreover, we may take the junior coefficients of the polynomials a1(x) and a2(x) to be unit, if a1(x),a1(x)≠0. If one of the polynomials a1(x),a1(x) is identical zero, then we may take the junior coefficients of the nonzero underdiagonal entries of the matrix A(x) to be unit. Such matrix A(x) we shall call thereduced matrix. All subsequent semiscalarly equivalent transformations of the matrix A(x) should not violate her property to be reduced.


## 4. Invariants of the Reduced Matrix

Theorem 8.
In reduced matrixA(x) of the form (2) codeg⁡a1, codeg⁡a2, and codeg⁡a3are invariants with respect to semiscalarly equivalent transformations.Proof.
LetA(x) and B(x) be reduced matrices of the form (2) and A(x)≈B(x). From equality (3), where matrix sij13 by Proposition 2 is upper triangular, we get (26)s22a1x+s23a3x≡s11b1x+s12a1xb1x+s13a3xb1xmodxk1.Recall that s11,s22≠0. If b1(x)≡0, then from (26) it follows that a1(x)≡0, i.e., codeg⁡a1=codeg⁡b1=+∞. Let a1(x)≠0 (b1(x)≠0). If codeg⁡a1<codeg⁡a3, then from (26) at once we have codeg⁡a1=codeg⁡b1. If codeg⁡a1>codeg⁡a3, then codeg⁡a3<k1<codeg⁡a2. In view of Proposition 3, we get (27)k1>codeg⁡a3=codeg⁡a2x,a3x,xk2=codeg⁡b2x,b3x,xk2=codeg⁡b3.Also by Proposition 3 we have codeg⁡(a1(x),a3(x),xk1)=codeg⁡(b1(x),b3(x),xk1). For this reason codeg⁡b1>codeg⁡b3=codeg⁡a3 and from (26) it follows that s23=0. Thus, codeg⁡b1=codeg⁡a1.
From equality (37) we can write(28)s33a2x≡b3xr12x+b2xr22xmodxk2,(29)s33a3x≡b3xr11x+b2xr21xmodxk2.We recall that s33≠0. If b2(x)≡0, then (29) implies that codeg⁡a3=codeg⁡b3. Then, from (28) we find a2(x)≡0, since r12(x)=s12xk1+s13a2(x), codeg⁡a2≠codeg⁡a3+k1.
Leta2(x)≠0 (b2(x)≠0). If codeg⁡b3<codeg⁡b2, then (29) implies that codeg⁡a3=codeg⁡b3, and from (28), taking into account the form of r12(x) and codeg⁡a3≠codeg⁡a2′, we have codeg⁡a2=codeg⁡b2.
Ifcodeg⁡b3≥codeg⁡b2, then from (28) we get also codeg⁡a2=codeg⁡b2. If 2codeg⁡b1<k1, then from (26) we obtain s12=0, a1(x)=b1(x) and r21(x)=0. Therefore, from (29) it is clear that codeg⁡a3=codeg⁡b3. In particular, a3(x)=0⇔b3(x)=0. If 2codeg⁡b1≥k1, then from (26) we have a1(x)=b1(x). In this case, rewrite (29) in the detailed form as (30)s33a3x-s11b3x+s12a1xδBx+s13a3xδBx-s23a3xb2′x≡0modxk2,where δB(x)=b1(x)b2′(x)-b3(x), b2(x)=xk1b2′(x). Since codeg⁡δB=codeg⁡(b1(x)b2′(x))=codeg⁡(a1(x)a2′(x)) and codeg⁡a3,codeg⁡b3≠2codeg⁡a1+codeg⁡a2′(x), as seen from the last congruence, codeg⁡a3=codeg⁡b3. Moreover, s12=0, if codeg⁡a3>2codeg⁡a1+codeg⁡a2′(x). Also in this case from last congruence we obtain a3(x)=0⇔b3(x)=0. Theorem is proved.Corollary 9.
Let in the reduced matrixA(x) one of the following three conditions hold true:(31)codeg⁡a3<codeg⁡a1≠+∞,(32)codeg⁡a3<codeg⁡a2′≠+∞,(33)+∞≠codeg⁡a3>2codeg⁡a1+codeg⁡a2′,codeg⁡a3≥codeg⁡a2.Then left reducible matrix in the passage from A(x) to the reduced matrix B(x) is of the form(34)s11s12s130s22000s33,if condition (31) is fulfilled, or (35)s110s130s22s2300s33,if one from two conditions (32), (33) is valid.Corollary 10.
Identical equality to zero of the entrya1(x), a2(x), or a3(x) of the reduced matrix A(x) is invariant with respect to semiscalarly equivalent transformations.Remark. If some two underdiagonal entries in the reduced semiscalarly equivalent matrices A(x), B(x) 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 matricesA(x), B(x) of the form (2) be given. Henceforth we shall apply the following notations:(36)ΔAx≔a1xxk1a3xa2x,ΔBx≔b1xxk1b3xb2xδAx=ΔAxxk1∈Cx,δBx=ΔBxxk1∈Cx.Corollary 11.
Ifa1(x),a2(x)≠0 and codeg⁡a3≥codeg⁡a2 in the reduced matrix A(x), then codeg⁡ΔA=codeg⁡(a1(x)a2(x)). Therefore, codeg⁡ΔA is an invariant with respect to semiscalarly equivalent transformations.Theorem 12.
In the reduced matrixA(x) the quantity codeg⁡ΔA is an invariant with respect to semiscalarly equivalent transformations.Proof.
LetA(x), B(x) be reduced matrices of the form (2) and A(x)≈B(x). Then from equality (3), where matrix sij13 is upper triangular (see Remark), we obtain(37)a2x=b2x-s12ΔBx-s13a2xδBx+s23a2xb2′x+xk2r32x,(38)a3x=a1xb2′x-δBx-s12a1xδBx--s13a3xδBx+s23a3xb2′x+xk2r31x,where b2′(x)=b2(x)/xk1∈C[x]. Excluding from (37) and (38) the summand, which contains s12, we define (39)ΔAx=ΔBx-ΔAxs13δBx-s23b2′x+xk2a1xr32x-xk1r31x.Since codeg⁡ΔA<k2+codeg⁡a1≤codeg⁡(xk2(a1(x)r32(x)-xk1r31(x)), codeg⁡ΔA<codeg⁡(ΔA(x)(s13δB(x)-s23b2′(x))), from (39) it follows that codeg⁡ΔA=codeg⁡ΔB.Corollary 13.
The congruencecodeg⁡ΔA≡0(modxk2+codega1) is an invariant of A(x) with respect to semiscalarly equivalent transformations.



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