Green's Function Theory

Lecture 26

We shall now direct our efforts towards finding what in linear algebra corresponds to the inverse of the linear operator $A-\lambda B$ . This means that we are going to find a linear operator $G$ which satisfies the equation

$$(A-\lambda B)G=I ~~.$$

Once we have found this inverse operator $G$ , it is easy to solve the inhomogeneous problem

$$(A-\lambda B)\vec x=\vec b$$

for $\vec x$ . This is so because the solution is simply

$$\vec x=G\vec b ~~.$$

If the vector space arena is an infinite-dimensional Hilbert space, the inverse operator

$$G_\lambda =(A-\lambda B)^{-1}$$

is usually called the Green's function of $A-\lambda B$ , although in the context of integral equations the expression

$$G_\lambda =(A-\lambda I)^{-1}$$

is sometimes called the resolvent of $A$ . Its singularities yield the eigenvalues of $A$ , while integration in the complex $\lambda$ -plane yields, as we shall see, the corresponding eigenvectors. It is therefore difficult to overstate the importance of the operator $G_\lambda$ .