Polynomial Bezoutian matrix with respect to a general basis

More on polynomial Bezoutians with respect to a general basis

“…On the other hand, Bezoutians have many applications in the theory of equations, system and control theory, etc., we refer the reader to the survey article of Helmke and Fuhrmann [10] and the book of Barnett [1] and the references therein. Recently the (classical) Bezoutian has been generalized to some other forms, in which the polynomial Bezoutian is an important direction of the research (e.g., see [3,4,7,12,13,14,18,19]). At the same time we have observed that in the recent work of Helmke and Fuhrmann [10], Fuhrmann and Datta [6], Mani and Hartwig [13], and Yang [18], etc., some properties of Bezoutians and their relation to system theoretic problems were derived by using operator approach and viewing the Bezoutian as a matrix representation of a certain operator in the dual bases.…”

“…Recently the (classical) Bezoutian has been generalized to some other forms, in which the polynomial Bezoutian is an important direction of the research (e.g., see [3,4,7,12,13,14,18,19]). At the same time we have observed that in the recent work of Helmke and Fuhrmann [10], Fuhrmann and Datta [6], Mani and Hartwig [13], and Yang [18], etc., some properties of Bezoutians and their relation to system theoretic problems were derived by using operator approach and viewing the Bezoutian as a matrix representation of a certain operator in the dual bases. While in the book of Heinig and Rost [9], a comprehensive discussion for the properties of classical Bezoutians was presented by using the methods of generating function and matrix algebra.…”

“…where are called the symmetrizer [11] and the generalized symmetrizer [18] of p(x) with respect to π(x) and Q(x), respectively. It can be seen that S Q (p) is (left) upper triangular and is congruent to S(p) (see Lemma 2.2 below).…”

“…In Section 2 we use the pure algebraic methods to re-derive three main results in [18] for polynomial Bezoutians; in contrast to operator approaches, are proofs are simpler and easier. Section 3 is devoted to investigation of some other properties of polynomial Bezoutians, they are mainly the generalizations of some results in [9].…”

“…The congruence relationship in Lemma 2.2 implies that B Q (p, q) has the same inertia or signature as B(p, q). Therefore the classical Hermite-Fujiwara and Routh-Hurwitz inertia and stability criteria can be generalized to the case of To this end we pure use algebraic methods to derive afresh three main results in [18] obtained by the third author. These results are the generalized Barnett formula, the intertwining relation between polynomial Bezoutian and confederate matrix, and the generalized Bezoutian reduction via polynomial Vandermonde matrix.…”