The main definitions of this chapter (those of the terms “eigenvalue” and “eigenvector”) come in §63. Once the basic theory associated with these terms has been developed, we shall be equipped to tackle effectively some questions whose importance was glimpsed in the course of chapter 7: e.g., for a given linear transformation a of a nonzero f.d. vector space V, is there a basis L of V such that M (a; L) is a diagonal matrix, and, if so, how can we find such a basis of V? These questions and related questions about matrices come to the forefront of our attention in §§65 and 66. In all this work, we shall make extensive use of what are called characteristic polynomials—a topic discussed as a preliminary in §62. The last two sections of the chapter (§§67, 68) break new ground in the theory of complex matrices.