ABSTRACT
Linear algebra over the field of real numbers is not always adequate. For example, we already had to deal with complex numbers while looking for the eigenvalues of the matrix
A =
(−1 2 −1 1
) ,
which are the roots of its characteristic polynomial χ A
(λ) = λ2 + 1. These are clearly a pair of imaginary numbers i and −i. In general, when A is real, its characteristic polynomial has real coefficients. Since the imaginary roots of polynomials with real coefficients occur in pairs, so do the eigenvectors belonging to imaginary eigenvalues. The extra effort involved in working with complex numbers is compensated somewhat by the fact that one needs to find only one member of a pair of complex eigenvectors. The other is given by its complex conjugate. For this chapter it is necessary to have a functional knowledge of complex numbers which, for the reader’s convenience, we summarize here.
