In this chapter we exclusively consider vector spaces over the field of reals unless otherwise stated. We first present a general discussion on bilinear and quadratic forms and their matrix representations. We also show how a symmetric bilinear form may be uniquely represented by a self-adjoint mapping. We then establish the main spectrum theorem for self-adjoint mappings based on a proof of the existence of an eigenvalue using calculus. We next focus on characterizing the positive definiteness of self-adjoint mappings. After these we study the commutativity of self-adjoint mappings. In the last section we show the effectiveness of using self-adjoint mappings in computing the norm of a mapping between different spaces and in the formalism of least squares approximations.

Bilinear and quadratic forms

Let U be a finite-dimensional vector space over ℝ. The simplest real-valued functions over U are linear functions, which are also called functionals earlier and have been studied. The next simplest real-valued functions to be studied are bilinear forms whose definition is given as follows.

Definition 5.1 A function f : U × U → ℝ is called a bilinear form if it satisfies, for any u, υ, w ∈ U and a ∈ ℝ, the following conditions.

1. f(u + υ, w) = f(u, w) + f(υ, w), f(au, υ) = af(u, υ).

2. f(u, υ + w) = f(u, υ) + f(u, w), f(u, aυ) = af(u, υ).

Let B = {u1, …, un} be a basis of U. For u, υ ∈ U with coordinate vectors x = (x1, …, xn)t, y = (y1, …, yn)t ∈ ℝn with respect to B, we have

where A = (aij) = (f(ui, uj)) ∈ ℝ(n, n) is referred to as the matrix representation of the bilinear form f with respect to the basis B.