Complex quadratic forms and self-adjoint mappings

Abstract

In this chapter we extend our study on real quadratic forms and self-adjoint mappings to the complex situation. We begin with a discussion on the complex version of bilinear forms and the Hermitian structures. We will relate the Hermitian structure of a bilinear form with representing it by a unique self-adjoint mapping. Then we establish the main spectrum theorem for self-adjoint mappings. We next focus again on the positive definiteness of self-adjoint mappings. We explore the commutativity of self-adjoint mappings and apply it to obtain the main spectrum theorem for normal mappings. We also show how to use self-adjoint mappings to study a mapping between two spaces. Complex sesquilinear and associated quadratic forms Let U be a finite-dimensional vector space over ℂ. Extending the standard Hermitian scalar product over ℂ n , we may formulate the notion of a complex ‘bilinear’ form as follows. Definition 6.1 A complex-valued function f : U × U → ℂ is called a sesquilinear form , which is also sometimes loosely referred to as a bilinear form , if it satisfies for any u , υ, w ∈ U and a ∈ ℂ the following conditions. f ( u + υ, w ) = f ( u, w ) + f (υ, w ), f ( u , υ + w ) = f ( u , υ) + f ( u, w ). f ( au , υ) = ā f ( u , υ), f ( u , a υ) = af ( u , υ). As in the real situation, we may consider how to use a matrix to represent a sesquilinear form. To this end, let B = { u 1 , …, u n } be a basis of U . For u , υ ∈ U , let x = ( x 1 , …, x n ) t , y = ( y 1 , …, y n ) t ∈ ℂ n denote the coordinate vectors of u , υ with respect to the basis B .