Matrix Representation Of The Factorization Of Greatest Common Divisor In The Framework Of Conjugate Product

Determine the greatest common divisor(GCD) of Polynomials is a fundamental problem in polynomial theory, and has widespread application in linear systems theory and network theory, so it has attracted a lot of attention in recent years. The computation of GCD is significant in the field of mathematics and control theory.In this dissertation, we investigate the factorization of a greatest common right(left) divisor. The matrix representation of extraction of greatest common right(left) divisor is proposed. The definition of right and left con-Sylvester matrices and con-Toeplitz matrices of a set of polynomials in the framework of conjugate product are firstly proposed. The relation between the factorization of right(left) con-Sylvester matrix and the extraction of common right(left) divisor is established. It is shown that the generalized con-Sylvester matrices may be factorized into the product of reduced con-Sylvester matrices and the con-Toeplitz matrices which represent the extracted common divisors. Then some properties of coprimeness of two polynomials are proposed. As the conjugate product is much regular than regular product, these properties also hold in the framework of regular product. After this, the properties of the rank of right and left con-Sylvester matrices are investigated. It is shown that the rank of right(left) con-Sylvester matrix is decided by the degree of greatest common right(left) divisor and the degree of the original polynomial set. Lastly, a matrix algorithm is proposed to calculate the right(left) common divisor. This algorithm is then verified by an example.