THE VANDERMONDE MATRIX IN THE GENERAL CASE

In an arbitrary complex Banach algebra, the Vandermonde matrix is considered. With the help of the accompanying Frobenius matrix, a connection is established between the coefficients of the algebraic equation and the Vandermonde matrix constructed from the roots, a definition of a divided difference of arbitrary order is given based on the invertible Vandermonde matrix. An analogue of the Hermite formula of the integral representation of the divided difference is given. An inclusion for the spectrum of the divided difference and an analogue of Dunford’s theorem on the mapping of spectra are given.