Companion matrix, Vandermonde matrix, Jordan form, Interpolating Polynomials, and Linear Transformations

Let F be a field, and let C be the n n companion matrix of the monic polynomial f (x) Fx such that f (x) = (xI-C) = (x - ₁) ^n₁ (x - ₘ) ^nₘ for m distinct elements ₁, , ₘ F. It is shown that there is a generalized Vandermonde matrix V associated with f (x) such that VCV^-1 is in Jordan form, and the columns of V^-1 are connected to the Hermite interpolating polynomials, whose higher derivatives will have specific values at ₁, , ₘ. If m = n and n₁ = = nₘ = 1, then the results reduce to the fact that the (classical) Vandermonde V of ₁, , ₙ satisfies VCV^-1 is a diagonal matrix and that the columns of V^-1 correspond to the Lagrange interpolating polynomials. This shows that the results for real polynomials and matrices also hold for polynomials and matrices over an arbitrary field F. Moreover, interpretations and insights of the results are given in terms of linear transformation between Fⁿ and the linear space of polynomials in Fx with degree less than n.