Two Matrix Systems Derived from a Polynomial of Even Order with Real Coefficients.

摘要

A polynomial f(z) of even order n can be written either in summation form with real coefficients AO, A1, ..., An, or in product form where the factors are quadratic polynomials in z. The coefficients of the factor polynomials are yv associated with z1 and xv associated with zO. A comparison of the coefficients in both forms yields systems of equations that can be systematically ordered. When xv is replaced by x, and yv by y, a Z matrix can be recognized as the essential part of one of the systems. Based on fundamental theorems of algebra, the Z matrix has been developed for polynomials of orders 2, 4, and 6. The Z matrix of order n has a strong internal construction. The elements of the Z matrix are simultaneous polynomials in x and y. the Y matrix can be derived from comparison of the coefficients or from the Z matrix. The Y matrix are polynomials in x only. The Z matrix and Y matrix are thus known and can be computed for a polynomial of any even order with real coefficients.