Numerical Methods in Computational Finance. Daniel J. Duffy
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Название: Numerical Methods in Computational Finance

Автор: Daniel J. Duffy

Издательство: John Wiley & Sons Limited

Жанр: Ценные бумаги, инвестиции

Серия:

isbn: 9781119719724

isbn:

СКАЧАТЬ 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)

      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 6 when 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 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.

      (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 normal lamda 1 x 1 plus normal lamda 2 x 2 right-parenthesis equals alpha left-parenthesis beta left-parenthesis normal lamda 1 x 1 plus normal lamda 2 x 2 right-parenthesis right-parenthesis 2nd Row equals alpha left-parenthesis normal lamda 1 beta x 1 plus normal lamda 2 alpha beta x 2 right-parenthesis equals normal lamda 1 alpha beta x 1 plus normal lamda 2 beta x 2 equals normal lamda 1 left-parenthesis alpha dot beta right-parenthesis x 1 plus normal lamda 2 left-parenthesis alpha dot beta right-parenthesis x 2 3rd Row Thus normal gamma left-parenthesis normal lamda 1 x 1 plus normal lamda 2 x 2 right-parenthesis equals normal lamda 1 normal gamma x 1 plus normal lamda 2 normal gamma x 2 4th Row for normal lamda 1 comma normal lamda 2 element-of upper K comma x 1 comma x 2 element-of upper W period EndLayout