In this paper the author discusses the simple properties of companion matrix of algebraic polynomial "P_n(x)=x~n-a_1x~(n-1)-a_2x~(n-2)-…-a_(n-1)x-a_n",and gives some general formulas of combinatorial identity and recurrent sequence such as "S(n)=a·S(n-1)+b·S(n-2)+c·S(n-3)" by using the eigenvalues of the matrix.
This paper discusses the inverse eigenvalue problem of companion matrix asfollow:give any sequences λ1,λ2, …, λn, then companion matrix M has λ1,λ2, …,λn, as its eigenvalues. The paper also points out the method to get the solution.
A new method to judge the relatively prime on double polynomials(a(λ)) and b(λ) is given in terms of the Barnett factorition of Bezout matrix and the unanimity of the zero-points with the eigenvalues of the first companion matrix of a(λ).
Balancing transformation on the companion matrix can greatly reduce its numerical illconditioned characteristics including huge Euclid norm and great differences among matrix elements. A numerical example validates the importance of balancing transformation technique.
Through the relationship of the real polynormial of degree n and an n×n companion matrix, this paper turns the question of solving for the roots of a polynormial into the question of finding out the eigenvalues of the matrix. By use of the power method to the companion matrix, this. paper has derived the Bernoulli formula and its procedure of deformation easily, thus the quicker formula of the power method can be used to the Bernoulli procedures to quicken the convergent speed.
An algorithm for power of companion matrix is presented in this paper. The algorithm described will be found to have practical application to computers for finding the region of zero of incomplete polynomials in the complex plane.The representation of the unique solution X of the matrix cquation AX-XB = E is also devoloped.