Resultant Matrices With Respect To A Special Type Of Polynomial Basis

The resultant of two polynomials is equal to the determinant of a resultant matrix, the entries of which are functions of their coefficients. There exist several different resultant matrices, and they may be considered equivalent because their rank deficiency is equal to the degree of the greatest common divisor (GCD) of the polynomials, and the coefficients of the GCD can be obtained by performing an LU or QR decomposition on the resultant matrix.The first chapter reviews over the background of the resultant matrices and the main work of this thesis. The second chapter is used to investigation of the companion matrix under the special type of polynomial basis {αi(n)(λ)=(1-λ)n-i(1+λ)i} and the inclusion regions for the eigenvalues of companion matrix. In the last part of this chapter, the greatest common divisor of two polynomials is determined by constructing a polynomial in a companion matrix. In Chapter 3, the first section discusses a generalized Bezout matrix under special bases. The generalized Bezout matrix is expressed by its displacement of the structure and the triangle decomposition in the second section.In Chapter 4, two properties of a companion matrix of non-derogatory and the intertwining relationship with symmetrizer are discussed in the first.And then in the second section, the relationship between companion resultant matrix and the Bezout resultant matrix is given and the Barnett’s factorization formula is extended.