Resultant matrices and Bezoutians under the polynomial bases generated by a bilinear transformation function

WU Hua-zhang

(School of Mathematical Sciences, Anhui University, Hefei 230601, China)



Resultant matrices and Bezoutians under the polynomial bases generated by a bilinear transformation function

WU Hua-zhang

(School of Mathematical Sciences, Anhui University, Hefei 230601, China)

bilinear transformation function; companion matrix; resultant matrix; Bezout matrix; Barnett’s formula

0 Introduction

whichestablishatransformationbetweentheupperhalfandthelowerhalfcomplexplanes.Ontheotherhand,thematrixcorrespondingtothebilineartransformationfunctionaboveisaMobiustransformationmatrixwhichcanbeusedtoestablishaone-to-onecorrespondencebetweenHankelandToeplitzmatrices[5].

Itiswellknown[6]thatthereexistcloserelationsbetweenresultantmatrixandBezoutmatrix(orsimplyBezoutian).TheBezoutianshavemanyimportantapplicationsinnumericalcomputing,controltheory,systemidentification,andnetworksandsignalprocessing.WerefertothebooksofBarnett[7],LancasterandTismenetsky[8],andBiniandPan[9].AmongthemethodsforthestudyofBezoutiansthepowerbasisisusuallygeneralizedtopolynomialcases.Ithasbeenshownin[10-11]thatmostpropertiesofclassicalBezoutian,suchastheBarnett-typeformula,theintertwiningrelationandthediagonalreduction,keepstillthesimilarformsforsomegeneralizedpolynomialBezoutians.

1 Resultant matrices

Thus, E-1Acanberegardedasacompanionmatrixofp(λ).Thepurposeofconstructingsuchacompanionmatrixismainlyfortheconvenienceofthecomputationoftheelements,notmerelyforitstheoreticalapplications.

whichimplies

(1)

Nowweconsidertheresultantmatrixoftwopolynomials.Set

(1) g(Cα)isaresultantmatrixoff(z)andg(z).

(2)ThedegreeoftheGCDd(z)off(z)andg(z)isequalton-rank(g(Cα)).

(3)Thecoefficientsofd(z)areproportionaltothelastrowofg(Cα)afterbeingreducedtorowechelonform.

Forexample,let

Thenthecompanionmatrixoff(x)is

thus rank (g(Cα))=1, and the degree of the GCD off(x) andg(x) is equal to 3-1=2. According to Theorem 1(3),d(x) should be proportional to

which is equal tog(x).

can be determined by

2 Bezoutians

(2)

For the computation of the elements of generalized Bezoutian, by [15] we have following formulas and ommit its proof. Here we still distinguish two different bases.

Thus the cost of the algorithm iso(n2).

Thus the cost of the algorithm iso(n2).

Thus, by the definition (2), the last equality equals

3 Properties and relationships of resultant matrices and Bezoutians

3.1 Properties of resultant matrices

respectively,alsokeepthesimilarrelationshipswiththeirtransposes.

Theorem 6 Assume that notations are as above. Then the following relations are satisfied

(3)

(4)

Proof Eqs.(3) can be checked by using a similar method as that in [15]. Here we omit the proof.

Theorem 7 Assume that notations are as before. Then we have

3.2 Relationships between Bezoutians and resultant matrices

which implies

(5)

By those equalities we can generalize the Barnett’s formula

to a general form. SinceB(p,1) is a special Bezout matrix withq(z)=1, then

Thus, Eq.(5) implies

(6)

Theorem 8 Assume that notations are as before. Then the generalized Barnett’s formulas hold

ThegeneralizedBarnett’sformulasaboveimplythefollowingresult.

Corollary 3 Assume that notations are as before andJthe reverse unit matrix. Then

Proof We only verify the first equality, the second one can be similarly proved and omitted. In view of Prop.2.11 in [5], we have

whereCpdenotes the companion matrix ofp(z), andJis the reverse unit matrix. Multipling byBTandBfrom the left hand side and the right hand side on the last equality respectively, and using Eqs.(3) and (5), we have

whereJα=BJBT.Theproofiscompleted.

AnintertwiningrelationofthegeneralizedBezoutianswiththeresultantmatricesarefulfilled.

Theorem 9 Assume that notation are as before. Then we have

Proof We only prove the first equality. By [8], we have the following equality

Using Eqs.(3) and (5), the remain is merely some elementary calculation.

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双线性变换函数生成基下的结式矩阵和Bezout矩阵

吴化璋

(安徽大学 数学科学学院,安徽 合肥 230601)

双线性变换函数;友矩阵;结式矩阵;Bezout矩阵;Barnett公式

10.3969/j.issn.1000-2162.2015.06.001

Foundation item:Supported by the Natural Science Foundation of Anhui Province (1208085MA02)

O151 Document code:A Article ID:1000-2162(2015)06-0001-08

Received date:2015-02-16

Author’s brief:WU Hua-zhang (1966-), male, born in Quanjiao of Anhui Province, professor of Anhui University,tutor for postgraduate.