ABSTRACT

In Chapter 1 we learned that a natural basis for IR n is the set of eigenvectors of a self-adjoint matrix, and that relative to this basis the matrix operator is diagonal. Similarly, in Chapters 3 and 4 we learned that certain integral and differential operators have the same property, namely, that their eigenvectors form a complete, orthonormal basis for the appropriate space, and that relative to this basis, the operator is diagonal. Unfortunately, this property of eigenvalue-eigenvector pairs is not true in general, since the eigenvectors of an operator need not form a basis for the space and, for some operators, there need not be any eigenvectors at all.