A fundamental theorem on lambda-matrices with applications —II. Difference equations with constant coefficients

Abstract

The fundamental theorem of the title refers to a spectral resolution for the inverse of a lambda-matrix L(λ) = ∑ i=0 l A iλ i where the A i are n× n complex matrices and det A l ≠ 0. In this paper general solutions are formulated for difference equations of the form ∑ i=0 l A iu r + i = ƒ γ, r = 1, 2,… . The use of these solutions is illustrated i new proof of Franklin's results describing the sums of powers of the eigenvalues of L(λ) (the generalized Newton identities), and in obtaining convergence proofs for the application of Bernoulli's method to the solution of ∑ i=0 l A iS i = 0 for matrix S.