Self-adjoint transformations

Abstract This chapter combines material concerning quadratic forms with material from the previous chapter on diagonalization. Throughout, V is a finite dimensional vector space over ℝ or ℂ with an inner product <|>. The main goal in this chapter is to understand the nature of quadratic forms, symmetric bilinear forms and conjugate-symmetric sesquilinear forms on a finite dimensional inner product space V—in particular, how the form relates to the inner product on V. It turns out that the key to describing such a form is an associated linear transformation on V. The linear transformations here are of interest in their own right, and have the property of being self-adjoint (as defined below). They can be diagonalized using methods in the last chapter, and this diagonalization provides a complete description of the bilinear or sesquilinear form we are interested in.