Daniel J. Duffy - Numerical Methods in Computational Finance

A real square matrix Q is orthogonal if:

(5.17)

Orthogonal matrices are important in numerical linear algebra applications because of their numeric stability properties. Some application areas are matrix decomposition methods (Golub and van Loan (1996)) such as:

QR decomposition orthogonal, upper triangular.

Singular Value Decomposition (SVD) and V orthogonal, diagonal matrix.

Eigendecomposition symmetric, Q orthogonal, diagonal.

Polar decomposition where is orthogonal, symmetric positive-semidefinite.

A particular application area is solving overdetermined systems of linear equations and solving ill-posed linear systems .

This is another very important class of matrices. Matrices having this property are highly desirable in applications and algorithms. An matrix is positive definite if:

(5.18)

and positive semidefinite if:

(5.19)

Some necessary conditions for positive semidefiniteness are:

The diagonal elements of A must be positive.

A is positive definite if and only if all its eigenvalues are positive.

The element of A having the greatest absolute value must be on the diagonal of A.

An example of a positive definite matrix A is:

To prove this let . Then

We can take the square root of a positive definite matrix A , and it is a well-defined function. Furthermore, we can factor a positive definite matrix A into:

(5.20)

where L is a lower triangular matrix having positive values on its diagonal. Equation (5.20)is called the Cholesky decomposition for A .

A matrix A is non-negative (written ) if all its elements are real and non-negative. A matrix A is greater than a matrix B if is positive. These kinds of matrices have many applications, for example Markov chains and stochastic matrix theory. They are also relevant when we construct monotone finite difference schemes and M -matrices to approximate the solution of differential equations, as we shall see later in this book.

A matrix A is said to be reducible if there exists a permutation matrix P such that:

The matrix A is called irreducible if no such permutation matrix exists.

The matrix A is said to be diagonally dominant if:

(5.21)

Some results that are used in PDE applications are:

1 If A is strictly diagonally dominant, then it is invertible.

2 If A is irreducible and diagonally dominant and if for at least one j, then A is invertible.

We conclude this section with a list of matrices whose properties are defined by the signs of their off-diagonal elements.

A Metzler matrix A satisfies where

A Z - matrix is a negated Metzler matrix where .

An M - matrix is a Z -matrix with eigenvalues whose real parts are non-negative. An M -matrix can be expressed in the form . The scalar s is at least as large as the maximum of the moduli of the eigenvalues of B , and I is the identity matrix.

Some applications of M -matrices are from mathematics and economics, for example:

Establish bounds on eigenvalues.

Convergence criteria for iterative methods.

Discretisations of PDEs (for example, in combination with exponential fitting) and monotone finite difference schemes.

Finite Markov chains.

Population dynamics.

Finally, L - matrices are defined by:

The Cayley Transform refers to a group of related concepts. The relevance to our work is that it appears when proving the stability of finite difference schemes for two-factor option pricing problems as we shall discuss in Chapters 18, 22and 23. It has been applied in fixed-income pricing as discussed in Davidson and Levin (2014).