Trace and Determinant

Abstract

The first section of this chapter has an introductory character; it explains the principles we use to define the trace and determinant. The precise definitions are given in the next two sections where we also derive the first properties of the trace and determinant. In Section 4 the analyticity of det(I − λA) as a function of λ is proved. The main theorem is given in the sixth section and expresses trace and determinant in terms of the eigenvalues. Some technical results from complex function theory, which are used in the proof of the main theorem, are derived in Section 5. The connections with the classical Fredholm determinant are described in Section 7. The last section contains as a first application two completeness theorems for eigenvectors and generalized eigenvectors.