Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Notation and convention
- 1 Vector spaces
- 2 Linear mappings
- 3 Determinants
- 4 Scalar products
- 5 Real quadratic forms and self-adjoint mappings
- 6 Complex quadratic forms and self-adjoint mappings
- 7 Jordan decomposition
- 8 Selected topics
- 9 Excursion: Quantum mechanics in a nutshell
- Solutions to selected exercises
- Bibliographic notes
- References
- Index
6 - Complex quadratic forms and self-adjoint mappings
Published online by Cambridge University Press: 18 December 2014
- Frontmatter
- Dedication
- Contents
- Preface
- Notation and convention
- 1 Vector spaces
- 2 Linear mappings
- 3 Determinants
- 4 Scalar products
- 5 Real quadratic forms and self-adjoint mappings
- 6 Complex quadratic forms and self-adjoint mappings
- 7 Jordan decomposition
- 8 Selected topics
- 9 Excursion: Quantum mechanics in a nutshell
- Solutions to selected exercises
- Bibliographic notes
- References
- Index
Summary
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 ℂn, 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) = āf(u, υ), f(u, aυ) = 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 = {u1, …, un} be a basis of U. For u, υ ∈ U, let x = (x1, …, xn)t, y = (y1, …, yn)t ∈ ℂn denote the coordinate vectors of u, υ with respect to the basis B.
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- A Concise Text on Advanced Linear Algebra , pp. 180 - 204Publisher: Cambridge University PressPrint publication year: 2014