The Bezoutian and the algebraic Riccati equation
- 1 June 1984
- journal article
- research article
- Published by Taylor & Francis in Linear and Multilinear Algebra
- Vol. 15 (3-4) , 265-278
- https://doi.org/10.1080/03081088408817595
Abstract
The classical Bezoutian is a square matrix which counts the common zeros of two polynomials in the complex plane. The usual proofs of this property are based on connections between the Bezoutian and the Sylvester resultant matrix. These proofs do not make transparent the nature of the Bezoutian as a finite dimensional operator. This paper establishes that the Bezoutian is a solution of a suitable operator Riccati equation which makes evident the connections between the Bezoutian as an operator and the common zeros of polynomials. One application to the inversion of block Hankel (Toeplitz) matrices is given. Brief discussions of other Bezoutian operators are included. Apparently, even in the classical case the connection between the Bezoutian and the Riccati equation has not been studied previously.Keywords
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