Triangulation and Decomposition of Endomorphisms

Abstract

In Section 9.3 we saw that Hermitian and complex unitary transformations always admit an orthogonal basis of eigenvectors, with respect to which these mappings are represented by simple diagonal matrices. This chapter presses the attack to find out what in general can be said about an arbitrary endomorphism T of a finite-dimensional vector space over an abstract field k. We first establish an astounding property of the characteristic polynomial; this is the content of the Cayley-Hamilton Theorem (10-1). Next, using similar techniques, we show that T is representable at least by an upper triangular matrix, provided that the roots of its characteristic polynomial all lie in k. This leads to the introduction of so-called nilpotent mappings. Maintaining our previous assumption on the characteristic polynomial, we then show that T is expressible as the sum of a diagonal map (easily built out of its eigenvalues) and a nilpotent map. Finally, further analysis of nilpotent endomorphisms yields a special matrix representation called the Jordan normal form.