Advanced Linear Algebra, Lecture 4.3: Generalized eigenvectors

Advanced Linear Algebra, Lecture 4.3: Generalized eigenvectors

Throughout, assume that A:X→X is an endomorphism of an n-dimensional vector space, or equivalently, an n-by-n matrix, over an algebraically closed field K. Since the polynomial ring K[t] is a principal ideal domain (PID), every ideal is generated by a single element. This means that the set (ideal) of polynomials p(t) such that p(A)=0 contains only multiples of a single polynomial m(t), called the "minimal polynomial" of A. By the Cayley-Hamilton theorem, this divides the characteristic polynomial. When A has repeated eigenvalues and not a full set of eigenvectors, the minimal polynomial has repeated roots. In this case, the eigenvectors can be extended into a basis for X by including so-called "generalized eigenvectors". Ordinary eigenvectors are characterized by being in the nullspace of A-λI. Generalized eigenvectors are in the nullspace of (A-λI)^m for some positive integer m. We see several 2x2 and 3x3 examples in this lecture, and give a novel way to visualize the generalized eigenvectors. The proof of these being a basis will be done throughout several subsequent lectures.