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Polynomial Bezoutian Matrix with Respect to a General Basisୋ Z.H

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Polynomial Bezoutian Matrix with Respect to a General Basisୋ Z.H View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Linear Algebra and its Applications 331 (2001) 165–179 www.elsevier.com/locate/laa Polynomial Bezoutian matrix with respect to a general basisୋ Z.H. Yang Department of Mathematics, East Campus, China Agricultural University, P.O. Box 71, 100083 Beijing, People’s Republic of China Received 7 September 2000; accepted 25 January 2001 Submitted by R.A. Brualdi Abstract We introduce a class of so-called polynomial Bezoutian matrices. The operator represen- tation relative to a pair of dual bases and the generalized Barnett factorization formula for this kind of matrix are derived. An intertwining relation and generalized Bezoutian reduction via polynomial Vandermonde matrix are presented. © 2001 Elsevier Science Inc. All rights reserved. AMS classification: 15A57 Keywords: Polynomial Bezoutian; Vandermonde matrix; Barnett-type formula 1. Introduction Given a pair of polynomials p(z) and q(z) with degree p(z) = n and degree q(z) n, the (classical) Bezoutian of p(z) and q(z) is the bilinear form given by − − p(x)q(y) − p(y)q(x) n 1 n 1 π(x,y) = = b xiyj (1.1) x − y ij i=0 j=0 × = and Bezoutian matrix of p and q is defined by the n n symmetric matrix B(p, q) n−1 bij i,j=0. ୋ Partially supported by National Science Foundation of China. 0024-3795/01/$ - see front matter 2001 Elsevier Science Inc. All rights reserved. PII:S0024-3795(01)00282-8 166 Z.H. Yang / Linear Algebra and its Applications 331 (2001) 165–179 Let n−1 Bst = π(x) = 1,x,...,x (1.2) be the standard power basis of the linear space Cn[x] of complex polynomials with degree less than n and n n j j p(z) = pj z (pn =/ 0), q(z) = qj z . (1.3) j=0 j=0 Then Eq. (1.1) can be rewritten as matrix form by p(x)q(y) − p(y)q(x) π(x,y) = = π(x)B(p,q)π(y)T, (1.4) x − y and we have the well-known Barnett factorization formula [1] (see also [3,9]) B(p, q) = S(p)q(C(p)), (1.5) where p1 p2 ··· ··· pn p2 p3 ··· pn 0 S(p) = B(p, 1) = . (1.6) . pn 0 ··· ··· 0 and 01··· ··· 0 001··· 0 . = . .. .. C(p) (1.7) 00··· 01 − p0 − p1 ··· ··· −pn−1 pn pn pn stand for the “symmetrizer” [15] and the companion matrix of p(z), respectively. As a special case of (1.4) (that is, q(x) = 1) we have the difference quotient form p(x) − p(y) D (x, y) = = π(x)B(p,1)π(y)T = π(x)S(p)π(y)T. (1.8) p x − y If we denote by pn pn−1 ··· p1 p0 0 .. pn . p1 p0 .. .. .. . pn−1 . ··· 0 pn pn−1 p1 p0 S1 S2 S = = (1.9) qn qn−1 ··· q1 q0 0 S S 3 4 . q .. q q n 1 0 .. .. .. . qn−1 . 0 qn qn−1 ··· q1 q0 Z.H. Yang / Linear Algebra and its Applications 331 (2001) 165–179 167 the 2n × 2n Sylvester matrix associated with p(z) and q(z) in (1.3), then it can be easily shown that B(p, q) defined by (1.1) can also be given by = − B(p, q) S1S4 S3S2 J, (1.10) where J is a reverse unit matrix of order n with 1’s along the secondary diagonal and 0’s elsewhere (see (1.14)). Bezoutian has appeared in the literature for a long history. It has contact con- nections with the theory of equations, polynomial stability and system theory, etc. For a more detailed exposition we refer the reader to the survey article of Helmke and Fuhrmann [12] and the books of Barnett [3] and Heinig and Rost [13]. Here we emphasize its applications in polynomial stability theory. For example, let p(z) have n simple 0’s x1,...,xn. Then Bezout matrix B(p, q) can be reduced to a diagonal matrix by congruence [8], that is, T = = n V(p) B(p, q)V (p) D diag p (xj )q(xj ) j=1, (1.11) = i−1 n where V(p) xj i,j=1 is the classical Vandermonde matrix corresponding to p(z). Eq. (1.11) shows that Bezoutian matrix B(p, q) has the same signature or iner- tia as diagonal matrix D, which was applied in [8] to prove the important Lienard–Chipart and its equivalent criteria for polynomial stability. In the case when p(z) has multiple 0’s xi with multiplicities ni (i = 1,...,r,n1 + ···+nr = n), the Bezoutian matrix was reduced to a block diagonal form by direct computation and operator methods by Sansigre and Alvarez [19], Chen and Yang [6], respectively. That is, T = r V(p) B(p, q)V (p) diag Ripi (Ji)q(Ji) i=1, (1.12) where V(p)denotes the confluent Vandermonde matrix defined by V(x1) T = r = . V(p) col V(xi) i=1 . , (1.13) V(xr ) in which π(xi) π (xi) n −1 1 dj π(x ) i 1! V(x ) = col i = i j . j! dx = . j 0 (n −1) π i (xi) (ni − 1)! and 168 Z.H. Yang / Linear Algebra and its Applications 331 (2001) 165–179 xi 10··· 0 00··· 01 0 xi 1 ··· 0 00··· 10 . .. .. .. Ri = . .,Ji = . (1.14) . .. .. .. 10··· 0 0 . 1 00··· 0 xi stand for the rotation matrix and Jordan block of order ni × ni corresponding to xi, respectively. Hereafter col(ai) denotes a column vector with ai as components. Bezoutian matrix has been generalized in different ways, mainly in two directions. One way is to define directly the Bezoutian matrix in the same manner as in (1.10) and (1.9) by using Sylvester matrix. For example, in [9] the Sylvester matrix in (1.9) is extended to the conjugate-Sylvester (CS) matrix Sc by pn pn−1 ··· p1 p0 0 .. c(pn) . c(p1)c(p0) . .. n−2 . .. .. . c (pn−1) . n−1 n−1 ··· n−1 n−1 = 0 c (pn) c (pn−1) c (p1)c (p0) Sc ··· qn qn−1 q1 q0 0 . .. c(qn) . c(q1)c(q0) . .. n−2 . .. .. . c (qn−1) . n−1 n−1 n−1 n−1 0 c (qn) c (qn−1) ··· c (q1)c (q0) S S = 1 2 , (1.15) S3 S4 where cm(z) denotes complex conjugation m times of z. Note that the matrices S1, S2, S3 and S4 in (1.15) are so-called conjugate-Toeplitz matrices, that is, if S =[sij ],thensi+1,j+1 = c(sij ). This generalization of Toeplitz matrices is discussed in [4,10]. Associated with p(z) and q(z) two polynomials n n k k c(z) = ckz ,d(z)= dkz ,cn =/ 0, (1.16) k=0 k=0 are introduced in [9] by the equations m k c (pk−1dm−k − qk−1cm−k) = 0,m= 1, 2,...,n+ 1, (1.17) k=1 and m k c (pn+k−mdn−k+1 − qn+k−mcn−k+1) = 0,m= 1, 2,...,n. (1.18) k=1 Z.H. Yang / Linear Algebra and its Applications 331 (2001) 165–179 169 If the CS matrix defined as in (1.15) associated with c(z) and d(z) is denoted by Tc and partitioned as T1 T2 Tc = , (1.19) T3 T4 then the conjugate-Bezoutian (CB) matrix Bc(p, q) associated with p(z) and q(z) is defined by = n+1 T − n+1 T Bc(p, q) Jc T1 JS4J Jc T3 JS2J (1.20) and Bc(p, q) has the same form of expression as in (1.10) (see [9, Eqs. (2.10) and (2.11)]). The second approach is to expand the generating function in (1.1) under a more general basis instead of the standard basis π(x) in (1.2). For example, in [5] Cheby- shev–Bezoutian matrices BT (p, q) = (tij ) and BU (p, q) = (uij ) are discussed, which are defined by p(x)q(y) − p(y)q(x) π(x,y) = x − y n−1 = tij Ti (x)Tj (y) i,j=0 n−1 = uij Ui (x)Uj (y), (1.21) i,j=0 = n−1 = n−1 where T(x) Ti(x) i=0 and U(x) Ui(x) i=0 stand for the Chebyshev poly- nomials bases of the first and second kinds, respectively, that is, T0(x) = 1,T1(x) = x, Tk(x) = 2xTk−1(x) − Tk−2(x) (1.22) and U0(x) = 1,U1(x) = 2x, Uk(x) = 2xUk−1(x) − Uk−2(x). (1.23) In full generality, let {Qk(x)} be a sequence of polynomials with degree Qk(x) = k(k= 0, 1,...)and set Q(x) ={Q0(x), Q1(x), . , Qn−1(x)}. (1.24) Evidently, Q(x) is a basis of Cn[x], which is called a general polynomial basis and has been studied extensively by Barnett and his co-authors [2,3,5,16]. General poly- nomial bases such as Newton and Chebyshev polynomials, etc., appear frequently in approximation theory and interpolation problems. In the present paper, our idea is to expand the generating function in (1.1) and (1.4) under the general polynomial basis Q(x) and define the so-called polynomial or gen- = n−1 eralized Bezoutian matrix with respect to Q(x), denoted by BQ(p, q) cij i,j=0, that is, 170 Z.H.
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