Numerical Methods in Computational Finance. Daniel J. Duffy

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      The function is called a kernel.

5.4 FROM VECTOR SPACES TO MATRICES

We are almost finished with our introduction to vector spaces and linear transformations. We now discuss how matrices arise and their relationship with the current topics.

      We discuss the notion of a matrix for a linear transformation. To this end, consider the linear transformation where V and W are finite-dimensional vector spaces, and let and be bases in V and W, respectively. Then:

(5.10)

      for some scalars . We can represent these scalars in rectangular form which we call a matrix:

      (5.11)

In this case we speak of a square matrix when otherwise, it is called a rectangular matrix. In short, each linear transformation determines a unique matrix with respect to the basis functions. Conversely, every such matrix determines a unique linear transformation from V to W defined by Equation (5.10) and the following mapping for a general vector :

      (5.12)

      5.4.1 Some Examples

      We take simple two-dimensional problems to model reflection and (counterclockwise) rotation in the plane.

       Case 1:(5.3)

       Case 2:From (5.10) we getwhich allows us to find the matrix A.

       Case 3:The trick is to build the matrix column by column, from top to bottom.

5.5 FUNDAMENTAL MATRIX PROPERTIES

      We first give a short history of how matrices were discovered.

       The term matrix was introduced by the 19th-century English mathematician James Sylvester, but it was his friend the mathematician Arthur Cayley who developed the algebraic aspect of matrices in two papers in the 1850s. Cayley first applied them to the study of systems of linear equations, where they are still very useful. They are also important because, as Cayley recognised, certain sets of matrices form algebraic systems in which many of the ordinary laws of arithmetic (e.g., the associative and distributive laws) are valid but in which other laws (for example, the commutative law) are not valid. (Wikipedia)

We give a precise overview of the operations that can be performed on matrices. We mention that they subsume operations on one-dimensional vectors because the latter can be viewed as matrices with one column (or with one row, depending on the context).

      The notation and operations for vectors are:

      (5.13)

      and for rectangular matrices:

      (5.14)

      Some special cases of matrices are:

       Row matrix: has one row (row vector).

       Column matrix: has one column (column vector).

       Zero matrix: all entries have the value zero.

       Diagonal matrix: all entries zero except those on main diagonal.

       Identity matrix: diagonal matrix all of whose diagonal elements == 1.

      Matrix operations are:

      (5.15)

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