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Numerical Methods in Computational Finance. Daniel J. Duffy

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Numerical Methods in Computational Finance - Daniel J. Duffy

V semicolon upper W right-parenthesis"/>. Then dimension upper V equals r left-parenthesis upper T right-parenthesis plus n left-parenthesis upper T right-parenthesis

dimension upper V equals r left-parenthesis upper T right-parenthesis plus n left-parenthesis upper T right-parenthesis

We are now interested in determining in how far two vector spaces are ‘similar’ in some sense.

Theorem 4.4 Let upper T element-of upper L left-parenthesis upper V semicolon upper W right-parenthesis . Then

T is onto W (surjective) if and only if .

T is one-to-one (injective) if and only if .

Definition 4.10 A linear transformation upper T element-of upper L left-parenthesis upper V semicolon upper W right-parenthesis is called an isomorphism if it is both injective and surjective.

It is possible to form the sum of linear transformations and to compose linear transformations, and we discuss this topic in Chapter 5.

Eigenvalues (Characteristic Roots) and Eigenvectors (Characteristic Vectors)

Let upper T element-of upper L left-parenthesis upper V semicolon upper V right-parenthesis . Then a scalar normal lamda element-of upper K is called an eigenvalue of L if there exists a non-zero x element-of upper V left-parenthesis upper K right-parenthesis (called an eigenvector) such that:

(4.17) italic upper T x equals normal lamda x period

Given a non-zero normal lamda , the set of all x satisfying (4.17) forms a subspace of upper V left-parenthesis upper K right-parenthesis . 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.

4.6 SUMMARY AND CONCLUSIONS

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.

CHAPTER 5 Guide to Matrix Theory and Numerical Linear Algebra

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

Georg Polya

5.1 INTRODUCTION AND OBJECTIVES

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

5.2 FROM VECTOR SPACES TO MATRICES

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

5.2.1 Sums and Scalar Products of Linear Transformations

We discuss two binary operators on the set upper L left-parenthesis upper V semicolon upper W right-parenthesis 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 5.1 The sum of two linear transformations alpha comma beta element-of upper L left-parenthesis upper V semicolon upper W right-parenthesis is a mapping from V to W and is defined by:

(5.1) left-parenthesis alpha plus beta right-parenthesis left-parenthesis x right-parenthesis equals alpha left-parenthesis x right-parenthesis plus beta left-parenthesis x right-parenthesis for-all x element-of upper V period

Definition 5.2 The scalar product of a linear transformation alpha element-of upper L left-parenthesis upper V semicolon upper W right-parenthesis and a scalar normal lamda element-of upper K is defined by:

(5.2) left-parenthesis normal lamda alpha right-parenthesis left-parenthesis x right-parenthesis equals normal lamda alpha left-parenthesis x right-parenthesis for-all x element-of upper V period

Definition 5.3 Let alpha element-of upper L left-parenthesis upper V semicolon upper U right-parenthesis , beta element-of upper L left-parenthesis upper W semicolon upper V right-parenthesis . Then the composition of alpha and beta is defined by:

(5.3) StartLayout 1st Row left-parenthesis alpha dot beta right-parenthesis left-parenthesis x right-parenthesis equals alpha left-parenthesis beta left-parenthesis x right-parenthesis right-parenthesis for-all x element-of upper W 2nd Row gamma identical-to alpha dot beta period EndLayout

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

StartLayout 1st Row left-parenthesis alpha dot beta right-parenthesis left-parenthesis <hr><noindex><a href=

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