Daniel J. Duffy - Numerical Methods in Computational Finance

(4.17)

Given a non-zero , the set of all x satisfying (4.17)forms a subspace of . The set of all eigenvectors of T corresponding to the same eigenvalue, together with the zero vector, is called an eigenspace (or the characteristic space) of T associated with that eigenvalue.

Eigenvalues and eigenvectors are important in numerical linear algebra.

We have given a precise and compact introduction to finite-dimensional vector spaces and linear transformations between vector spaces. We introduce the notation and jargon associated with the topic, and it forms the basis for many applications. In particular, it clears the way for a study of matrix theory and numerical linear algebra.

We recommend Shilov (1977) as an elegant introduction to linear algebra.

If you can't solve a problem, then there is an easier problem you can solve: find it .

Georg Polya

The main goal of this chapter is to introduce matrices: what they are and how to create and use them, as well as classifying matrices based on some of their intrinsic and computed properties. This is not a book on matrix theory, but we think that it is important to introduce matrices upfront and not to relegate them to a two-page appendix at the end of the book. We prefer to inform the reader of the prerequisites in the first part of the book rather than at the end when all the other chapters have been discussed.

We continue with this topic in Chapter 6when we discuss the role of matrices in numerical linear algebra and their integration with finite difference schemes for ordinary differential equations.

We continue with the topics in Chapter 4and show how matrices are representations of linear operators.

We discuss two binary operators on the set where V and W are vector spaces, namely the sum of two linear transformations and multiplication of a linear transformation by a scalar. Each operator produces a new linear transformation.

Definition 5.1The sum of two linear transformations is a mapping from V to W and is defined by:

(5.1)

Definition 5.2The scalar product of a linear transformation and a scalar is defined by:

(5.2)

Definition 5.3Let , . Then the composition of and is defined by:

(5.3)

We check that the composition is a linear transformation as follows:

An inner product (a generalisation of dot product from high school calculus) on a real vector space V is a scalar-valued function on the Cartesian product of V with itself having the following axioms:

(5.4)

Inner products on complex vector spaces are also possible, but a discussion is outside the scope of this chapter. We also note that inner products are sometimes written as (for example, in physics) instead of where x and y are vectors. For the specific case of n -dimensional vectors we usually write the inner product as follows:

(5.5)

where is the transpose of vector .

This latter notation is common in linear algebra and applications. Of course, more general vector spaces will need the more generic form in axioms (5.4).

An inner product space is a vector space on which an inner product is defined. A finite-dimensional real inner product space is known as a Euclidean space , and a complex inner product space is known as a unitary space . The length of a vector x in Euclidean space is defined to be , and the angle between two vectors x and y is given by:

(5.6)