Linear Algebra with Applications, 10th Edition

Chapter 8 Canonical Forms

8.1 Nilpotent Operators

If a linear transformation L mapping an n-dimensional complex vector space into itself has n linearly independent eigenvectors, then the matrix representing L with respect to the basis of eigenvectors will be a diagonal matrix. In this chapter, we turn our attention to the case where L does not have enough linearly independent eigenvectors to span V. In this case, we would like to choose an ordered basis of V for which the corresponding matrix representation of L will be as nearly diagonal as possible. To simplify matters, in this first section, we will restrict ourselves to operators having a single eigenvalue $\lambda$ of multiplicity n. It will be shown that such an operator can be represented by ...