Decomposing C with the Adjoint Equation

Abstract

Before proceeding directly to the formal theory, let us attempt to motivate (as much as we can) the concepts which are going to be introduced. In the proof of Lemma 20.1, namely, formulas (20.4) and (20.5), we encountered a bilinear form in a very natural manner. In fact, if λ is a solution of (20.1), then formulas (20.3), (20.4), (20.5) show that the equation (A-λI)ϕ = ψ has a solution if and only if, for every nonzero row vector a for which aΔ(λ) = 0, it follows that (ae ^-λİ,ψ) = 0 where (α,ψ) is defined in (20.5). This certainly suggests an “alternative” theorem and indicates that the bilinear form (α,ψ) will be important in the theory. Also, it suggests that the “adjoint” operator of A should have eigenfunctions of the form e^−λθ, o ≤ θ ≤ r, so that the “alternative” theorem could be stated in the following manner: the equation (A-λI)ϕ = ψ has a solution if and only if (α,ψ) = o for all solutions α of the equation (A*-λI) α = 0.