if 5.6.6 Irreducible Matrices
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.
5.6.7 Other Kinds of Matrices
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
A Z-matrix is a negated Metzler matrix
An M-matrix is a Z-matrix with eigenvalues whose real parts are non-negative. An M-matrix can be expressed in the form
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:
5.7 THE CAYLEY TRANSFORM
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, 22 and 23. It has been applied in fixed-income pricing as discussed in Davidson and Levin (2014).
In general, we define:
(5.22)
In the case where A is positive definite, we can compute the norm of Q as follows:
This result was proved in Kellogg (1964), and it is an important lemma to prove unconditional stability of splitting schemes. In the same way it is possible to show that
Appendix : The Schrödinger Equation
We discuss the time-dependent one-dimensional Schrödinger PDE and its finite difference approximation by the Euler, fully implicit, and Crank–Nicolson schemes. We prove that only the Crank–Nicolson scheme is unitary (preserves norms) as discussed in Section 5.6.3 in the context of unitary matrices.
In
